text:
## Friday, February 14, 2014

### capturing stdout in a python child process

Lately at work I've been putting together a web management console that handles execution of some of our long running Gurobi models. This makes a lot of sense to do since the models run on Gurobi Cloud for hours or even days before achieving their desired optimality gaps. Gurobi Cloud requires a persistent connection, so we use a server to handle that instead of solving models from our desktop machines. In a sense, this is a bit of a throwback to the time sharing systems from before the PC era.

As a result, I've had to update my dormant (and, to be honest, never particularly acute) administration and systems programming skills. There are always some interesting gotchas that come up in this sort of work. This time I ran into the following conundrum: I have a cron job that is responsible for constructing model instances in a child process, handing them off to Gurobi Cloud, and saving their results. Its output is redirected to a file so I can tail it in real time and store it for later. For the web console, I also want to store the output to a database so I can easily display it. And I'd like to do this in as generic a manner as possible, so I can add other types of models (custom heuristics, for instance) and also capture anything they write to standard output and standard error.

There are a number of threads available on capturing standard output. Some of the advice I found recommended using os.dup2 to clone the stdout file descriptor into another file. The problem with this approach is it requires having an open file, which is inconvenient, and it ends up closing the existing stdout. So the output does get redirected, but then it disappears from my log. Also, I could only seem to capture it once. I wanted to do this repeatedly.

Other advice got me closer to the mark: Replace sys.stdout with an instance of StringIO, then just read its contents later. This allowed me to do redirections in serial, but once again meant that standard output was essentially shut off.

The final solution was to subclass StringIO and override its write method. Every time write is called, it writes to both the in-memory string and a canonical reference to stdout. Example code is given below that captures stdout and stderr from a child process while still letting their messages be visible.

from StringIO import StringIO
from multiprocessing import Process, Queue
import sys

# Proxy classes that both write to and capture strings
# sent to stdout or stderr, respectively.
class TeeOut(StringIO):
def write(self, s):
StringIO.write(self, s)
sys.__stdout__.write(s)

class TeeErr(StringIO):
def write(self, s):
StringIO.write(self, s)
sys.__stderr__.write(s)

class Parent(object):
def run(self):
# Start child process.
queue = Queue()
child = Process(target=self.spawn, args=(queue,))
child.start()
child.join()

# Do someting with out and err...
out = queue.get()
err = queue.get()

def spawn(self, queue):
# Save everything that would otherwise go to stdout.
out = TeeOut()
sys.stdout = out

err = TeeErr()
sys.stderr = err

try:
# Do something...
print 'out 1'
print 'out 2'
sys.stderr.write('error 1\n')
print 'out 3'
sys.stderr.write('error 2\n')

finally:
# Restore stdout, stderr and send their contents.
sys.stdout = sys.__stdout__
sys.stderr = sys.__stderr__
queue.put(out.getvalue())
queue.put(err.getvalue())

if __name__ == '__main__':
Parent().run()

A few minor points:

• StringIO can't be subclassed like object-based types, thus lines 9 and 14 can't use super.
• The sys module keeps two references to stdout: sys.stdout and sys.__stdout__. We use the latter to recover the former after processing.
• The contents of standard out and standard error are transmitted from the child process to its parent at the end of processing. A smarter implementation might transfer chunks of them periodically, but I'm not expecting them to be large.
• The transfer happens in a finally block. Thus any run time exception in the child will not interfere with logging.

## Monday, February 3, 2014

### chebyshev centers of polygons with gurobipy

A common problem in handling geometric data is determining the center of a given polygon. This is not quite so easy as it sounds as there is not a single definition of center that makes sense in all cases. For instance, sometimes computing the center of a polygon's bounding box may be sufficient. In some instances this may give a point on an edge (consider a right triangle). If the given polygon is non-convex, that point may not even be inside or on its boundary.

This post looks at computing Chebyshev centers for arbitrary convex polygons. We employ essentially the same model as in Boyd & Vandenberghe's Convex Optimization text, but using Gurobi instead of CVXOPT.

Consider a polygon defined by the intersection of a finite number of half-spaces, $Au \le b$. We assume we are given the set of vertices, $V$, in clockwise order around the polygon. $E$ is the set of edges connecting these vertices. Each edge in $E$ defines a boundary of the half-space $a_i'u \le b_i$.

$V = \{(1,1), (2,5), (5,4), (6,2), (4,1)\}$
$E = \{((1,1),(2,5)), ((2,5),(5,4)), ((5,4),(6,2)), ((6,2),(4,1)), ((4,1),(1,1))\}$

The Chebyshev center of this polygon is the center point $(x, y)$ of the maximum radius inscribed circle. That is, if we can find the largest circle that will fit inside our polygon without going outside its boundary, its center is the point we are looking for. Our decision variables are the center $(x, y)$ and the maximum inscribed radius, $r$.

In order to do this, we consider the edges independently. The long line segment below shows an arbitrary edge, $a_i'u \le b_i$. The short line connected to it is orthogonal in the direction $a$. $(x, y)$ satisfies the inequality.

The shortest distance from $(x, y)$ will be in the direction of $a$. We'll call this distance $r$. If we were to move the edge so it had the same slope but went through $(x, y)$, its distance from $a_i'u = b_i$ would be $r||a_i||_2$. Thus we can add a constraint of the form $a_i'u + r||a_i||_2 \le b_i$ for each edge and maximize the value of $r$ as our objective function.

max $r$
subject to $(y_i-y_j)x + (x_j-x_i)y + r\sqrt{(x_j-x_i)^2 + (y_j-y_i)^2} \le (y_i-y_j)x_i + (x_j-x_i)y_i\, \forall\, ((x_i,y_i), (x_j,y_j)) \in E$

As this is linear, we can solve it using any LP solver. The following code does so with Gurobi.

#!/usr/bin/env python
from gurobipy import Model, GRB
from math import sqrt

vertices = [(1,1), (2,5), (5,4), (6,2), (4,1)]
edges = zip(vertices, vertices[1:] + [vertices[0]])

m = Model()
m.update()

for (x1, y1), (x2, y2) in edges:
dx = x2 - x1
dy = y2 - y1
m.addConstr((dx*y - dy*x) + (r * sqrt(dx**2 + dy**2)) <= dx*y1 - dy*x1)

m.setObjective(r, GRB.MAXIMIZE)
m.optimize()

print 'r = %.04f' % r.x
print '(x, y) = (%.04f, %.04f)' % (x.x, y.x)

The model output shows our center and its maximum inscribed radius.

r = 1.7466
(x, y) = (3.2370, 2.7466)

Question for the reader: in certain circumstances, such as rectangles, the Chebyshev center is ambiguous. How might one get around this ambiguity?

## Monday, July 29, 2013

### network splitting

Last week, Paul Rubin wrote an excellent post on Extracting a Connected Graph from an existing graph. Lately I've been performing related functions on data from OpenStreetMap, though without access to a solver. In my case I'm taking in arbitrary network data and splitting it into disconnected subnetworks. I thought it might be a good case study to show an algorithmic way doing this and some of the performance issues I ran into.

A small example can be seen below (click on the image to enlarge it). This shows a road network around the Las Vegas strip. There is one main (weakly) connected network in black. The roads highlighted in red are disconnected from the main network. We want code that will split these into connected subnetworks.

Say we have data that looks like the following. Instead of nodes, the numbers in quotes represent edges. Think of these as streets.

{
"0": [1, 2, 3],
"1": [9248, 9249, 9250],
"2": [589, 9665, 9667],
"3": [0, 5, 6],
"4": [0, 5, 6],
"5": [588],
"6": [4, 8, 9],
...
}

Our basic strategy is the following:

1. Start with every edge alone in its own subnetwork.
2. For each connection, merge the networks of the source and destination edges.

#!/usr/bin/env python
import json
import sys
import time

class hset(set):
'''A hashable set. Note that it only hashes by the pointer, and not by the elements.'''
def __hash__(self):
return hash(id(self))

def __cmp__(self, other):
return cmp(id(self), id(other))

if __name__ == '__main__':
try:
inputfile = sys.argv[1]
except:
print 'usage: %s network.json' % sys.argv[0]
sys.exit()

print time.asctime(), 'parsing json input'

edge_to_net = {} # Edge ID -> set([edges that are in the same network])
nets = set()     # Set of known networks

print time.asctime(), 'detecting disconnected subgraphs'
for i, (from_edge, to_set) in enumerate(connections.iteritems()):
from_edge = int(from_edge)

try:
from_net = edge_to_net[from_edge]
except KeyError:
from_net = edge_to_net[from_edge] = hset([from_edge])

if not (i+1) % (25 * 1000):
print time.asctime(), '%d edges processed / %d current subnets' % (i+1, len(nets))

for to in to_set:
try:
to_net = edge_to_net[to]

# If we get here, merge the to_net into the from_net.
if to_net is not from_net:
to_net.update(from_net)
for e in from_net:
edge_to_net[e] = to_net
nets.remove(from_net)
from_net = to_net

except KeyError:
edge_to_net[to] = from_net

print time.asctime(), len(nets), 'subnets found'

We run this against the network pictured above and it works reasonably quickly, finishing in about 7 seconds:

Mon Jul 29 12:22:38 2013 parsing json input
Mon Jul 29 12:22:38 2013 detecting disconnected subgraphs
Mon Jul 29 12:22:38 2013 25000 edges processed / 1970 current subnets
Mon Jul 29 12:22:44 2013 50000 edges processed / 124 current subnets
Mon Jul 29 12:22:45 2013 60 subnets found

However, when run against a road network for an entire city, the process continues for several hours. What is the issue?

The inefficiency occurs from lines 46 to 50. In this we are frequently removing references to every element in a large set. Instead, it would be better to remove as few references as possible. Therefore, instead of merging from_net into to_net, we will determine which network is the smaller of the two and marge that one into the larger one. Note that this does not necessarily change the worst case time complexity of the algorithm, but it should make the code fast enough to be useful. The new version appears below.

#!/usr/bin/env python
import json
import sys
import time

class hset(set):
'''A hashable set. Note that it only hashes by the pointer, and not by the elements.'''
def __hash__(self):
return hash(id(self))

def __cmp__(self, other):
return cmp(id(self), id(other))

if __name__ == '__main__':
try:
inputfile = sys.argv[1]
except:
print 'usage: %s network.json' % sys.argv[0]
sys.exit()

print time.asctime(), 'parsing json input'

edge_to_net = {} # Edge ID -> set([edges that are in the same network])
nets = set()     # Set of known networks

print time.asctime(), 'detecting disconnected subgraphs'
for i, (from_edge, to_set) in enumerate(connections.iteritems()):
from_edge = int(from_edge)

try:
from_net = edge_to_net[from_edge]
except KeyError:
from_net = edge_to_net[from_edge] = hset([from_edge])

if not (i+1) % (25 * 1000):
print time.asctime(), '%d edges processed / %d current subnets' % (i+1, len(nets))

for to in to_set:
try:
to_net = edge_to_net[to]

# If we get here, merge the to_net into the from_net.
if to_net is not from_net:
# Update references to and remove the smaller set for speed.
if len(to_net) < len(from_net):
smaller, larger = to_net, from_net
else:
smaller, larger = from_net, to_net

larger.update(smaller)
for e in smaller:
edge_to_net[e] = larger
nets.remove(smaller)
edge_to_net[to] = larger
from_net = larger

except KeyError:
edge_to_net[to] = from_net

print time.asctime(), len(nets), 'subnets found'

Indeed, this is significantly faster. And on very large networks it runs in minutes instead of hours or days. On the small test case used for this post, it runs in under a second. While this could probably be done faster, that's actually good enough for right now.

Mon Jul 29 12:39:55 2013 parsing json input
Mon Jul 29 12:39:55 2013 detecting disconnected subgraphs
Mon Jul 29 12:39:55 2013 25000 edges processed / 1970 current subnets
Mon Jul 29 12:39:55 2013 50000 edges processed / 124 current subnets
Mon Jul 29 12:39:55 2013 60 subnets found

## Wednesday, July 17, 2013

### stack in go

I've been using an increasing amount of Go lately. One of my favorite features is its ability to add methods to any existing type. This is possible for primitive types by simply providing an alias for them, similar to using typedef in C.

Recently, I found I needed a stack container. Nothing complicated, merely an implementation of push, pop, and empty methods. While this is easy enough to do using slices, wrapping the required code makes for a much nicer interface.

package main

import "fmt"

type Element float64
type Stack []Element

func (s *Stack) Push(e Element) {
*s = append(*s, e)
}

func (s *Stack) Pop() Element {
e := (*s)[len(*s)-1]
*s = (*s)[:len(*s)-1]
return e
}

func (s *Stack) Empty() bool {
return len(*s) < 1
}

func main() {
s := Stack([]Element{})

for i := 0; i < 10; i++ {
s.Push(Element(i))
}

for !s.Empty() {
fmt.Println("pop:", s.Pop())
}
}

Since there aren't real classes in Go, or inheritance, this doesn't do anything scary like monkey patching. It's just a bit of syntactic sugar to make our lives better. For instance, the Push method could just as easily look like this to the compiler:

func Push(s *Stack, e Element) {
*s = append(*s, e)
}

## Monday, October 8, 2012

### web slides with python & math using deck.js, syntaxhighlighter, and mathjax

In putting together some slides for class that include both Python code and math models, I realized that it was high time to update the technology stack I'm using. If anyone else finds this useful, please feel free to copy my starting template and use or improve upon it.

I'm hosting my slides directly out of Dropbox since I can do so without uploading them to a web server. It also means I can edit and view them locally without have to go through any sort of manual syncing process. This is possibly the simplest way in existence to share a page on the internet.

Steps:

1. Create the directories /public/, /public/www/, /public/www/slides/, and /public/www/slides/boilerplate/ in your Dropbox folder.
3. Download SyntaxHighlighter and decompress it into /public/www/syntaxhighlighter/.
4. Download MathJax and decompress it into /public/www/mathjax/.
5. Save this template into /public/www/slides/boilerplate/boilerplate.html.

Now to create a new slide deck, copy the boilerplate directory to something new in /public/www/slides/. You should be able to begin with something that looks like this. Search through its source for "slides start" to see how to create slides that show beautifully rendered Python and LaTeX.

Use the left and right arrow keys or the icons that appear when you hover over the left and right of the slides to navigate.

## Monday, September 24, 2012

### roll your own gurobi shell

Note: These instructions have been updated to use the setup.py from the Gurobi distribution.

I've recently started using Gurobi for a class. While I'm pleased with the level of Python support they provide, I'm not a fan of the Gurobi shell they ship it with.

• The shell is Python, which is good. However, instead of bundling the Gurobi libraries as a Python module that I can install alongside my system's native Python, they ship their own Python binaries. If I want to use any third-party modules I have to make them available to Gurobi somehow.
• They appear to have removed distutils from the Python they ship.
• The Gurobi shell doesn't give me module docs, tab completion, or anything I'd get from either ipython or bpython. It's nothing more than the rudimentary cPython shell.

They do, however, provide a setup.py script, so we don't have to use gurobi.sh or the Python that's in the distribution. Here are simple instructions for using bpython as your shell. You should have Gurobi 5.0.1 installed and the GUROBI_HOME environment variable set. (I assume you are installing it into your home directory.)

$mkdir -p$HOME/lib/python2.7/site-packages/
$cd$GUROBI_HOME
$PYTHONPATH=$HOME/lib/python2.7/site-packages/ python setup.py install --prefix=$HOME  Update your environment's PYTHONPATH and LD_LIBRARY_PATH: $ echo "export PYTHONPATH=\$PYTHONPATH:\$HOME/lib/python2.7/site-packages/" >> $HOME/.bashrc$ echo "export LD_LIBRARY_PATH=\$LD_LIBRARY_PATH:\$GUROBI_HOME/lib/" >> $HOME/.bashrc$ source $HOME/.bashrc  Install bpython: $ sudo apt-get install bpython

Now type bpython and you should have the lovely interface you see below, complete with tab completion and module docs. Isn't that better?

## Saturday, September 22, 2012

### lagrangian relaxation with gurobipy

Note: this is still a new technique to me, so please point out any errors and I will correct them in the post.

Correction: $b_{ij}$ is now $b_j$ in the second set of constraints.

We've been studying Lagrangian Relaxation (LR) in the combinatorial optimization course I'm taking this term, and I had some difficulty finding a simple example covering its application. In case anyone else finds it useful, I'm posting a Python version for solving the Generalized Assignment Problem (GAP). This won't discuss the theory of LR at all, just give example code using Gurobi 5.0.1.

The GAP as defined by Wolsey consists of a maximization problem subject to a set of set packing constraints followed by a set of knapsack constraints:

max $\sum_i \sum_j c_{ij} x_{ij}$
subject to $\sum_j x_{ij} \leq 1 \forall i$
$\sum_i a_{ij} x_{ij} \leq b_{ij} \forall j$
$x_{ij} \in \{0, 1\}$

An unadulterated version of this BIP using gurobipy might look like the following:

#!/usr/bin/env python

# This is the GAP per Wolsey, pg 182

from gurobipy import GRB, Model

model = Model('GAP per Wolsey')
model.modelSense = GRB.MAXIMIZE
model.setParam('OutputFlag', False) # turns off solver chatter

b = [15, 15, 15]
c = [
[ 6, 10,  1],
[12, 12,  5],
[15,  4,  3],
[10,  3,  9],
[ 8,  9,  5]
]
a = [
[ 5,  7,  2],
[14,  8,  7],
[10,  6, 12],
[ 8,  4, 15],
[ 6, 12,  5]
]

# x[i][j] = 1 if i is assigned to j
x = []
for i in range(len(c)):
x_i = []
for j in c[i]:
x.append(x_i)

# We have to update the model so it knows about new variables
model.update()

# sum j: x_ij <= 1 for all i
for x_i in x:

# sum i: a_ij * x_ij <= b[j] for all j
for j in range(len(b)):
model.addConstr(sum(a[i][j] * x[i][j] for i in range(len(x))) <= b[j])

# max sum i,j: c_ij * x_ij
model.setObjective(
sum(
sum(c_ij * x_ij for c_ij, x_ij in zip(c_i, x_i))
for c_i, x_i in zip(c, x)
)
)
model.optimize()

# Pull objective and variable values out of model
print 'objective =', model.objVal
print 'x = ['
for x_i in x:
print '   ', [1 if x_ij.x >= 0.5 else 0 for x_ij in x_i]
print ']'

The solver quickly finds the following optimal solution of this toy problem:

objective = 46.0
x = [
[0, 1, 0]
[0, 1, 0]
[1, 0, 0]
[0, 0, 1]
[0, 0, 0]
]

There are two sets of constraints we can dualize. It can be beneficial to apply Lagrangian Relaxation against problems composed of knapsack constraints, so we will dualize the set packing ones. The first thing we do is replace the constraints generated in lines 39-40 with a new set of variables, penalties, which take the values of the slacks on the set packing constraints. We then modify the objective function, adding Lagrangian multipliers times these penalties.

Instead of optimizing once, we do so iteratively. An important consideration is we may get nothing more than a dual bound from this process. Any integer solution is not guaranteed to be primal feasible unless it satisfies complementary slackness conditions -- for each dualized constraint either its multiplier or penalty must be zero.

Setting the intial multiplier values to 2 and using subgradient optimization with a step size of 1 / (iteration #) to adjust them, we arrive at the following model:

#!/usr/bin/env python

# This is the GAP per Wolsey, pg 182, using Lagrangian Relaxation

from gurobipy import GRB, Model

model = Model('GAP per Wolsey with Lagrangian Relaxation')
model.modelSense = GRB.MAXIMIZE
model.setParam('OutputFlag', False) # turns off solver chatter

b = [15, 15, 15]
c = [
[ 6, 10,  1],
[12, 12,  5],
[15,  4,  3],
[10,  3,  9],
[ 8,  9,  5]
]
a = [
[ 5,  7,  2],
[14,  8,  7],
[10,  6, 12],
[ 8,  4, 15],
[ 6, 12,  5]
]

# x[i][j] = 1 if i is assigned to j
x = []
for i in range(len(c)):
x_i = []
for j in c[i]:
x.append(x_i)

# As stated, the GAP has these following constraints. We dualize these into
# penalties instead, using variables so we can easily extract their values.
penalties = [model.addVar() for _ in x]
model.update()

# Dualized constraints: sum j: x_ij <= 1 for all i
for p, x_i in zip(penalties, x):
model.addConstr(p == 1 - sum(x_i))

# sum i: a_ij * x_ij <= b[j] for all j
for j in range(len(b)):
model.addConstr(sum(a[i][j] * x[i][j] for i in range(len(x))) <= b[j])

# u[i] = Lagrangian Multiplier for the set packing contraint i
u = [2.0] * len(x)

# Re-optimize until either we have run a certain number of iterations
# or complementary slackness conditions apply.
for k in range(1, 101):
# max sum i,j: c_ij * x_ij
model.setObjective(
sum(
# Original objective function
sum(c_ij * x_ij for c_ij, x_ij in zip(c_i, x_i))
for c_i, x_i in zip(c, x)
) + sum (
# Penalties for dualized constraints
u_j * p_j for u_j, p_j in zip(u, penalties)
)
)
model.optimize()

print 'iteration', k, 'obj =', model.objVal, \
'u =', u, 'penalties =', [p.x for p in penalties]

# Test for complementary slackness
stop = True
eps = 10e-6
for u_i, p_i in zip(u, penalties):
if abs(u_i) > eps and abs(p_i.x) > eps:
stop = False
break

if stop:
print 'primal feasible & optimal'
break

else:
s = 1.0 / k
for i in range(len(x)):
u[i] = max(u[i] - s*(penalties[i].x), 0.0)

# Pull objective and variable values out of model
print 'objective =', model.objVal
print 'x = ['
for x_i in x:
print '   ', [1 if x_ij.x >= 0.5 else 0 for x_ij in x_i]
print ']'

On this problem LR converges very quickly to an optimal solution, as we can see here:

iteration 1 obj = 48.0 u = [2.0, 2.0, 2.0, 2.0, 2.0] penalties = [0.0, 0.0, 0.0, 0.0, 1.0]
iteration 2 obj = 47.0 u = [2.0, 2.0, 2.0, 2.0, 1.0] penalties = [0.0, 0.0, 0.0, 0.0, 1.0]
iteration 3 obj = 46.5 u = [2.0, 2.0, 2.0, 2.0, 0.5] penalties = [0.0, 0.0, 0.0, 0.0, 1.0]
iteration 4 obj = 46.1666666667 u = [2.0, 2.0, 2.0, 2.0, 0.16666666666666669] penalties = [0.0, 0.0, 0.0, 0.0, 1.0]
iteration 5 obj = 46.0 u = [2.0, 2.0, 2.0, 2.0, 0.0] penalties = [0.0, 0.0, 0.0, 0.0, 1.0]
primal feasible & optimal
objective = 46.0
x = [
[0, 1, 0]
[0, 1, 0]
[1, 0, 0]
[0, 0, 1]
[0, 0, 0]
]

Exercise for the reader: change the script to dualize the knapsack constraints instead of the set packing constraints. What is the result of this change in terms of convergence?

## Thursday, March 29, 2012

### python-zibopt 0.7.2 released

python-zibopt 0.7.2 has been released and the big news is that it supports both Python 2 and 3. (Thanks to Guillaume Sagnol for the patch!)

Other changes include:

• Added magic square and normal magic square examples
• Fixed issue with single-variable objectives subject to constraints:
solver += x >= y
solution = solver.minimize(objective=x)
• Constant objectives no longer raise errors:
solver.minimize(objective=3)
• Fixed obscure constraint issue:
solver += x <= (y-1) + 100*(1-z)

From here I'm hoping to focus my efforts on the Cython port, which should give a better basis for more advanced functionality in future versions.

## Tuesday, March 27, 2012

### spring coding haiku

Spring priorities
ever shift as with the winds.
Effort flows downstream.

## Monday, March 26, 2012

### a few thoughts on music and computer programming

Greg Linch started a discussion recently on Twitter regarding music and coding. The apex of the conversation was Zed Shaw pointing to his own well-stated comments on the matter.

I got back just over a couple hours ago from a weekly rehearsal, so I've got a number of things I'd like to mention swimming around in my head. Here are a few:

1. To "Linguistic Skills" I'd add that musical notation is specifically built to enable certain domain-specific elements. Its staved nature allows a high degree of parallelism. I can't think of any other common language that succinctly expresses multiple concurrent paths in a readable format. One can glean a great deal of information about independently moving pieces from something like an orchestral score very quickly, in ways that we cannot with programming or other languages. Thinking concurrently is increasingly important in our contemporary computing environment.

2. Musical notation is evolving rapidly. If one studies modern music, one will be constantly presented with new types and forms of instruction. In a way, this is like introducing new op-codes or programming constructs and can help form a desirable malleability of mind. Consider the notation, for example, in Penderecki's "Threnody for the Victims of Hiroshima" or much crazier looking things by composers such as Xenakis. Sometimes learning a single piece requires understanding the structure of an entirely new language.

3. Music encourages team playing. In a chamber ensemble, one is expected to become so in sync with the other members that you do things like breathe together. A well-rehearsed group will practice such things as playing notes slowly to try and achieve better temperament (consistent intonation which should result in more commonly placed overtones) and vibratos that waver at the same rates. Communication is rehearsed in the form of breaths, nods, eye contact, and other physical gestures (chamber players cannot over-communicate). Most chamber music involves a lot of passing back and forth of musical phrases, which can be analogous to message passing in a synchronous computing system.

4. Interpretation of a piece is largely subjective and therefore requires lots of compromise in order to become cohesive. This sort of compromise doesn't just mean allowing someone else to have his or her way, but means accepting and buying into it once a decision is made. If a member of my group plays a passage a certain way, I'm now expected to play that passage in a similar manner. Once a design decision is agreed upon, the impetus is on the players to accept and use it to improve the overall structure.


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## Crew Rotation In the Superyacht Industry – Part I

Although commercial shipping has rotated crew for many years, it is a recent phenomenon in the superyacht industry. It started to become a reality on yachts first with engineers, and then on the larger yachts where manning regulations required officers with STCW qualifications.

When rotation first started is a little unclear; it was certainly in use in the late 70’s when I was with BP Shipping – though it only applied to deck and engineering officers. However, what was clear, that it was not born out of regulation, but driven by market forces and a recognition that to attract and keep the right people they had to offer a better work/life balance. Today, these very same reasons are relevant to yachting.

So, in our latest post, we are going to take a dive into rotation and, due to its importance to many captains, their families and yacht owners, we will break it down into two parts and focus on the whys, how’s and the impact this fundamental employment change has had on the yachting community.

This week in Part I, Brendan is writing of his own employment journey to supply some context, then next week in Part II, Malcolm will take a deeper look at the pros and cons and how you might present the idea to a yacht owner using a worked example.

### Brendan

I entered yachting when contracts and structured leave were rare to the point of not known. Leave was when it suited the yacht’s programme, often with little notice. Crew would scramble to make last minute arrangements when a window opened. With limited leave, weekends in port was our time to get away, explore and socialise, leaving the yacht to the care of a couple of watchkeepers. Regardless of age or relationship status, life revolved around the yacht and no alternate lives ashore were maintained.

This began to change for me in 2007 when I joined my first 100m+ yacht and realised there was no stopping a yacht of this scale. It needed crew every day to keep the show running and there were no more ‘weekends with a couple of watchkeepers’. I remember naively saying to the Heads of Departments we would shut the yacht down one weekend for everyone’s rest; they humoured me, said yes, but ignored the instruction and kept the yacht working the way it demanded.

Since that time, I have modelled many and various employment structures for yacht owners and their representatives. When I do these, I do not speak of rotation from a crew’s perspective, it is with consideration of the yacht owners needs and their investment. My point being that the yacht, the owner and guest experience should not suffer because of crew taking leave. I support this position with a crude calculation; add the finance cost of the yacht to the operating expenses and divide by 365 to gain the cost per day of the yacht’s existence. The number can be staggering and to think that you would intentionally stop the operation so the team can take days off does not show good value.

Further to this crude calculation, the owner is reminded that the beauty of yacht ownership is freedom and spontaneity. Rotation can allow that when a gap opens in their diary, they can escape to their yacht and enjoy the pleasure of being on the water with family and friends; something that is even more relevant today.

In one of the presentations the yacht owner agreed for senior crew without hesitation, saying, “but yes, they have families, and we want them to be focussed on us when they are here and not worried about when they can get home.” For junior crew there was a different perspective with the principal asking, “why do these crew want so much time away when they were young?” Weren’t these the years to earn money, travel and gain experience needed to progress?

Malcolm’s comment – the latter point I also heard from an owner. One 80m+ example lost several junior crew because of generous leave/rotation! The basic reasoning was it was expensive to spend so much time at home, all their friends were working so no one to hang out with, and it took far longer to gain the necessary sea time and experience to progress. Sometimes you cannot win!

### Be careful what you wish for.

Since 2007 (outside of shipyard construction) I have been on equal time rotation. This is a Nirvana for many but, having defended the position to the owner that the yacht requires 365-day attendance from its captain and senior team to get rotation over the threshold, you are accountable to work accordingly.

So now, during a 2-3-month roster onboard, I tend to focus completely on the yacht and my days exploring the wonderful areas I sail through are a distant memory. Crew come and go in and procession of rotational changes and although bonds are still made perhaps, they are a little weaker? That said, when they return refreshed, the faces are familiar and they quickly adapt back to life onboard without missing a beat; ensuring operational readiness, a consistent service quality, better maintenance and safety.

It could be said that with better leave and rotation means the yacht is now the place we work; it is no longer the centre of our universe and the place where we also lived our lives!

Clearly this is a much healthier balance but, occasionally, I do look back on those time long ago in sepia, when spending 11-months of the year with the same tight crew created my most memorable experiences and learning opportunities. I am open in saying my memory is grander than the reality, it was unsustainable if I wanted any sort of normal life outside of yachting. I could not have raised a family without rotation and so today I am content with a few laps around the yacht at anchor or a quick morning run on the rare times in port. My days exploring are not lost, I now have the time and freedom to return to destinations in my own time and with my family, and that is incomparable.

Done right, better leave and rotation offer crew and yacht owners many benefits and, although there are added costs, carefully planned, they are not as high as might be imagined, and there are many advantages that cannot be measured purely in monetary terms that can add value to the yacht owning experience.

In Part II Malcolm’s deep dive is where you need to go to look to the tools that you might need when structuring your own rotational plans to a yacht owner, their representative or yacht manager. The strength of your case will depend not only of the financial model, but also the quality of your reasoning and supporting facts. Without a compelling case, the yacht owner or their representatives might be thinking “living the dream, sailing the seas, working half a year and still complaining?

By |2020-11-05T12:55:57+01:00October 22nd, 2020|captains, regulations, yachtowners|

## COVD-19 Recommendations From the Healthy Sail Panel

It could be suggested there is too much information available on COVID-19 and the pandemic; including, an almost infinite number of articles and commentary on the internet, numerous Circulars and Guidance from the IMO and, publications from the International Chamber of Shipping (ICS) such as Coronavirus (COVID-19) Guidance for Ship Operators for the Protection of the Health of Seafarers (version 3.0 29th September 2020).

This excess of information can be confusing and, also as suggested in Tom Nichols book “The Death of Expertise” result in a tendency to trust in the internet to make us ‘experts’ in all manner of subjects and, resist or even ignore, advice from those with a deep understanding and experience of the subject matter, including COVID-19 – this can lead to poor outcomes.

With the absence of a collective response from the industry, it has been left to individual yacht management companies and/or captains and crew to wade through the mass of information, try to assess its quality and efficacy, and then develop and implement their own protocols and procedures in response to the virus. And, whilst some of these are well thought out and effective, others on deeper analysis, are perhaps like the ‘Swiss Cheese’ risk assessment analogy, have holes for the virus to pass through.

### Recommendations from the Healthy Sail Panel

So, it was a great relief to come across the “Recommendations from the Healthy Sail Panel.” This is the first document I have seen from a related industry with a well-researched and holistic approach to the prevention, protection and mitigation of COVID-19, in an easy to follow format.

The Healthy Sail Panel is a collaboration between Royal Caribbean Group and Norwegian Cruise Line Holdings Ltd who put together a panel of World leading experts to help inform and find a new pathway back to the “new normal” of sailing. The resulting research and recommendations are broken down into 5 key focus areas, with over 70 recommendations, many of which are applicable to yachting.

The key focus areas are: –

1. Testing, Screening and Exposure Reduction
2. Sanitation and Ventilation
3. Response, Contingency Planning and Execution
4. Destination and Excursion Planning
5. Mitigating Risks for Crew Members

COVID testing is one of the subjects with references in the footnotes. Further reading clearly highlights the value of testing for screening and diagnosis but, like the use of electronic aids to navigation, you have to be aware of the limitations, errors and accuracy.

I was certainly confused by the various tests; Rapid Antigen, PCR, Antibody, etc., their effectiveness for screening, diagnosing present and past COVID infection. The US CDC footnote reference in the Healthy Sail Panel certainly helped my understanding, along with the infographic below – found on the Nature website in their article Fast coronavirus tests: what they can and can’t do.”

##### Courtesy: Nature “Fast coronavirus tests: what they can and can’t do”

It became clear that, amongst other factors, the timeline of infection has a big effect on the various tests and why caution is required – especially with the Rapid Antigen tests that may be used by yacht crew.

### Indian Ocean and Caribbean Passage

As many yachts and crew are readying themselves for passages to the Caribbean, Indian Ocean or further afield, I thought it was also worth considering this in the context of COVID-19 and posing the following question: –

“Should you self-isolate the yacht and crew and test before departing?”

Clearly, the time taken in transit is likely a suitable quarantine period for destination arrival purposes. However, the reason I pose the question is that given that most yachts will be departing from countries/areas with high rates of infection, and crew will have been enjoying shore leave and their time in port, what happens if a crewmember is infected, but tests negative (if tested) and is asymptomatic prior to departure?

Once underway and symptoms present, not only would there be concerns of further infections amongst the crew, and medical treatment if severely affected, there would also be concerns about at the port of destination; would the port allow the yacht to berth and what are the reception and medical facilities for any infected crew?

The same goes for ‘crossing crew’ do you bring them in early and quarantine (onboard in single cabin) and test prior to departure?

Clearly, no captain wants to restrict well-earned shore leave but, then again, it is important to avoid any crewmember being infected and becoming a medical emergency and/or a vector for further transmission, especially on a long sea voyage, so it makes sense to try and prevent this outcome.

I’d be interested to know how yachts and management companies are dealing with this. Some considerations:

• What methods are in use for mitigating the risk of infection prior to departure
• Has the port authority of your arrival destination been contacted and what is their policy in the event of an infected crewmember on an arriving yacht
• Do the hospitals have the facilities and capacity to handle a COVID-19 patient
• Are there any additional medical supplies and PPE above ‘Medical Scales’ that may be recommended to carry

The above, departs slightly from the main reason for this post but, for those about to embark on a lengthy passage, it’s something worth thinking about?

As always at OnlyCaptains, or goal is to share knowledge and help inform. Hopefully, the Healthy Sail Panel offers some useful information on COVID-19 that may help with your own procedures. And, perhaps it might be used as a reference by industry associations such as MYBA, LYBRA, IYBA, in a collaborative effort to create our own yachting recommendations. These would not only be of value to captains, crew and yacht management, they would also help to instil confidence in owners and charterers through the knowledge that industry accredited measures were in place to protect them whilst onboard.

By |2021-06-14T09:38:08+02:00October 9th, 2020|captains, covid-19, environment, yachtowners|

## Measuring Yacht Efficiency – How And Why It Matters

Last week, I attended the Yacht Cub de Monaco: Capital of Yachting Experience. It was a very well organised and attended event, with some very interesting presentations and discussions.

It was also the launch of the Yacht Club de Monaco Superyacht Eco Association (SEA) INDEX. Supported by Nobiskrug and Credit Suisse, this is an important initiative with a goal to benchmark yachts in terms of their CO2 environmental performance. And, whilst there are other emissions, CO2 is by far the largest greenhouse gas (GHG) of importance and the one most visible in the public eye.

The principle is that it uses the IMO’s Energy Efficiency Design Index (EEDI) formula with a few changes to make it more specific to yachts.

The SEA INDEX is the first tool designed to assess and compare the efficiency of a yachts design and its environmental impact in terms of CO2, with a transparent and easy to understand rating system. Stars are awarded from 1 (lowest rating) to 5 (highest rating) depending on where a yacht sits above or below the rating bands relative to the baseline of sampled yachts.

Image: Courtesy of the SEA INDEX

The data from approx. 130 yachts of various length and displacement was sampled and their data entered in order to develop the initial baseline – there are now over 200 yachts.

It uses max power and speed, which may seem excessive, but a metric was required and, if you consider this as the ‘maximum emissions potential’ of a yacht, by using the same set of data points for all yachts, it provides a ‘standard’ for comparing their designs. For example, on comparable sized yachts, a more efficient hull will require less power for the same speed, and more efficient HVAC and hotel systems power management, will require smaller generators, both of which will result in reduced emissions and a higher INDEX rating.

And, as new designs and engineering innovations are introduced into yachting, the SEA INDEX will help highlight the improvements being made.

Of course, actual emissions depend on many variables that are affected by an owner and the operational profile of a yacht – these are hard to assess in any consistent or meaningful way. If we had recorded all yacht activity and consumption over the last 10 or 20 years, we would be able to draw a curve of standard deviation and have an idea of what might be described as ‘average use’ on which to make comparisons. Unfortunately, we don’t have this information, and this is perhaps the flaw in all such tools, so the only true account of a yachts CO2 emissions has to be calculated from their fuel consumed.

The factor the IMO use for CO2 emissions from MGO is 3.206, this means for every 1,000t of MGO used, 3,206t of CO2 is generated so it is easy to calculate your CO2 from fuel.

Any design efficiency gains, and improvements that can be made in the operation of the yacht, such as running at lower speed, managing power, switching off unused lighting and equipment, etc., will reduce the power required, fuel consumed and emissions.

In combination with efficiency gains, Carbon offsetting is one way to mitigate a yachts emission. Though, as I have written in a previous piece Superyacht Carbon Offsetting great care is required to select one that is fit for purpose.

But, it’s not just the amount of CO2 that is an important consideration. Looking to the future, it is very likely that shipping, like other industries, will be impacted by Market Based Mechanism’s (MBM) to drive forward the transition to a greener future, and these will have cost implications.

The IMO by 2023 will introduce their new framework for the reduction of GHG emissions from shipping and it could include a carbon tax. The EU in a recent plenary session of parliament, agreed that shipping should be included in the EU Emissions Trading Scheme (ETS) possibly in 2021and include vessels less than 5000gt. Trafigura, one of the World’s largest ship charterers, published on 25th September “A proposal for an IMO-led global shipping industry decarbonisation programme” calling for a $100 –$200 tax per ton of CO2 on shipping as the only way of driving the necessary industry change.

As further evidence of the direction of travel for CO2 emissions for business, Swiss Re made this announcement on the 15th September 2020:

“Swiss Re steps up its internal carbon levy to USD 100 per tonne as of 2021 and will gradually increase it to USD 200 per tonne by 2030”

Any such taxes or levies imposed on CO2 emissions will increase the cost of yacht ownership.

On top of that we have Environmental Governance and Sustainability (EGS) targets that are becoming ever more prevalent, especially in investment and finance. The Poseidon Principles is just one initiative, launched the 18th June 2019, “major shipping banks will for the first time integrate climate considerations into lending decisions to incentivize maritime shipping’s decarbonization” their goal is to work towards the IMO 2030 and 2050 reductions in GHG by ensuring that their loan books are aligned with those targets – finance will become harder for vessels that fail to meet efficiency improvements and GHG reductions.

Could similar lending rules apply to yachts in the future, how would that affect the value of older less efficient yachts?

Whilst it is not yet clear how taxes and regulations will be imposed in the future, what is clear, is that yachting is unlikely to escape their embrace. And our intimate connection to the sea and the environment places additional responsibility on the industry to protect the health of our oceans and planet. The SEA INDEX is the first of many important tools, including those from the Water Revolution Foundation, that will help us to understand our environmental footprint and drive the necessary change that puts us on a pathway to a sustainable superyacht industry.

Like any instrument that is reliant on data; the more yachts that participate, the more refined and accurate the SEA INDEX will become – I would call upon all Captains to get involved.

By |2021-06-14T09:38:51+02:00September 29th, 2020|captains, environment, regulations, yachtowners|

## Yacht and Commercial Qualifications and the Path to Yachts >3000gt

With the superyacht fleet >3000gt still growing, I thought it worthwhile to look at the difference between Yacht and Commercial qualifications, the various pathways to an Unlimited CoC and, some considerations to factor into your choice of pathway.

A Little Background

We know that education, training and experience are vital to ensuring safety on yachts and ships, and although there is the Standards of Training Certification and Watchkeeping (STCW) Convention that regulates minimum standards, it was introduced for commercial ships and those destined for a merchant navy career.

In recognition of the requirements of the yachting industry, in 2002 the MCA and, their contributing partners, introduced MGN 195(M) the first yacht specific standards of training and certification as allowed under Article IX of STCW.

Prior to this, commercial STCW qualifications or RYA qualifications for recreational boaters were the only qualifications available to professional yacht crew. Given yachting’s trajectory, it made absolute sense to combine elements of both to develop a qualification that was relevant and attainable. There is no doubt that the Large Yacht Code has significantly improved the standards of safety and professionalism within our industry.

Though there have been amendments, and we are now at MSN1858 (M+F), there are perhaps questions whether it has adapted sufficiently to the changes and challenges we have seen within our industry and, of course, why there is still a barrier that prevents Yacht captains serving on Yachts >3000gt? To answer this and other questions we need to explore the options available.

For unlimited tonnage qualifications under MSN1856 (M+F) there are several different pathways. If we look at the MCA Higher National Certificate/Diploma (HNC/HND) route for experienced mariners the one experienced yacht crew would be more inclined to choose – there are significant differences in the educational and sea service requirements. For example, the classroom time requires an extra 53 weeks compared to Yacht qualifications.

Another difference is bridge watchkeeping. No verifiable bridge watchkeeping duties or, experience, is required for a Yacht OOW/Chief Mate <3000gt, only 4 months for Master 500gt and, 8 months for a Master Yacht <3000gt and Marshall Islands (MI) Master (Yachts) Unlimited – this is in stark contrast to the minimum of 30 months required for a STCW Master Unlimited under MSN1856 (M+F).

Perhaps these are the reasons why it has proved so difficult for the MCA to provide a suitable transition – how do they reconcile the differences and still comply with STCW? Perhaps an Endorsement, like there are for special vessels like tankers, could be a way forward; though, it should be pointed out these Endorsements are on top of STCW qualifications.

The Marshall Islands recognised an opportunity to provide a pathway to unlimited tonnage yachts for experienced captains and introduced their own qualification. This allows experienced captains, through additional modules and an assessment by MI examiners, to obtain an Unlimited CoC restricted to Yachts.

Below is a summary of the main differences.

MCA Master Yachts <3000gt

• 21 weeks education and short courses
• Minimum 60 months sea service – including 8 months of bridge wathckeeping
• Cost approx. €25,000

MI Master (Yachts) Unlimited – including MCA Master Yachts <3000gt

• 31 weeks of education and short courses,
• Minimum 72 months sea-service – including, 8 months of bridge watchkeeping, and 12 months served as a Master on a yacht >500GT
• Cost approx. €40,000

MCA Master Unlimited – following the HNC/HND experienced seafarer route

• 61 (college) + 13 (home study) a total of 74 weeks of education and short courses
• Minimum 60 months sea-service, including 24 – 36 months of bridge watchkeeping, and 6 months watch keeping duties prior to OOW
• Cost approx. €25,000

Of course, there will be additional expense for a MCA Master Unlimited due the course duration, loss of earnings and subsistence costs.

From Yacht to Unlimited

There is the MI Master (Yachts) Unlimited and, although a practical option for those with Yacht qualifications to upgrade, there are some considerations. So far, the Cayman Islands are the only Administration to have recognised this CoC, the fleet of Yachts >3000gt is small, and the Yacht restriction, further reduces its value compared to an Unlimited commercial CoC.

Unfortunately, the MCA don’t make it easy. A Yacht captain, irrespective of experience, still must start at the beginning with an Unlimited OOW, followed by Chief Mate Unlimited before Master Unlimited. Due to time/cost considerations, this only makes sense for those who make the decision to follow the commercial pathway early in their career or, already have an STCW OOW or Chief Mate Unlimited, and wish to upgrade.

There are other Administrations, such as AMSA in my example below, who also have their own pathways. But these are generally based on STCW.

Interestingly, there is a growing number of deck crew, especially those working on the larger yachts, who are choosing the commercial route, it certainly provides them with better career opportunities, including commercial shipping, harbour Pilotage, ferries, as well as shore-based employment with a Class Society, Flag Administration or insurance company.

Whatever pathway you choose to Master Unlimited, there will be a cost and no guarantee of the returns. You might have to accept work as a 2nd officer or chief officer to gain the relevant large yacht experience. And, the reality is that, as mentioned, the Yacht >3000gt fleet is small so opportunities are limited, and you will be competing against commercial officers and captains who have now made yachting their home.

My Own Journey

Late in my career I embarked on my own journey to Master Unlimited. I had earned my OOW Unlimited working for BP Shipping before yachting and with that and my sea service on yachts (all Private) I was accepted into the Chief Mate Unlimited/Master Unlimited pathway with AMSA, the Australian Maritime Safety Agency.

The process involved attending college full-time for almost 10 months in Fremantle, Australia, for my Chief Mate Unlimited. This was an experience, as I was, let me say, a more mature student, and my classmates were all about 20 years younger, with relevant commercial experience; I was that ‘white boat captain’ a bit of a novelty! I was certainly self-conscious of that from the outset. However, what became clear was that, as seafarers, ultimately, we face the same maritime challenges, and whether a ‘white boat’, box boat, tanker, or anchor handler, there was much similarity and we all benefited from the different experiences we brought to the classroom.

On obtaining my Chief Mate Unlimited I then returned to sea to earn another 12 months of sea service necessary to sit my Master Unlimited orals exam. This was not easy as AMSA required sea service on vessels over 3000gt for Master Unlimited (they were supposed to reduce this to 500gt but Australian politics intervened). Fortunately, my industry connections came to the rescue and I found myself as Chief Officer on Katara then Eclipse where I obtained the bulk of the sea service required. I then returned to Fremantle for a months prep before successfully passing the oral exam and becoming Master Unlimited.

So, starting with an OOW Unlimited to becoming a Master Unlimited, took nearly 3 years and I had already worked for nearly 20 years as a yacht captain. And, the the cost of taking a significant amount of time off, leaving a good job with loss of earnings, living expenses, college fees, etc., all mounted up.

Summing Up

So how was my own return on investment?

If I’m brutally honest, the financial rewards have not been great. However, for me, the sense of achievement, professional pride, my greater knowledge and understanding, and the experiences gained from my time served on Yachts >3000gt, have been a far more important measure of success than the monetary cost.

For those starting their career, it is worth considering the MCA or, equivalent, Commercial qualification pathway. This will provide you with the greatest flexibility and broader range of opportunities in the future. And not just at sea, there are many related industries who value a Master Unlimited qualification.

Unfortunately, until the MCA change their qualification standards and pathways, the choice is limited for experienced yacht officers/captains who want to serve on Yachts >3000g. The options are, the MI Master Yachts (Unlimited) or, the MCA Unlimited or, other Administrations, Commercial pathways.

Finally, it’s worth remembering that there are some amazing yacht Owners and yachts below 3000gt and that bigger, is not always betterthey come with challenges and differences that may not suit. Think carefully about the time and cost involved and your career goals. And, find time to speak to those who have already trodden the path

By |2021-06-14T09:40:10+02:00August 27th, 2020|captains, qualifications|

## The Captain And Yacht Owner Relationship

It is often overlooked, but the most important relationship for a successful yachting experience is that between the Captain and the Yacht Owner, or Principal Charterer. Long past the heady days when deals are signed and photos are taken with designers, brokers and shipyard owners cutting ribbons, the Captain and the Crew are tasked with delivering on the promise.

I was recently asked a question during an interview, “As a Superyacht Captain, how do you manage the expectations of an Owner?”

On the face of it, a simple question but the answer is rather more complex. Let’s think of a sample 100 metre yacht; the cost may exceed €200M and the yacht owner has waited 5 years – 2 years of development with designers and brokers, and 3 years in construction. So when their dream is finally delivered and their anticipation is heightened, how does a Superyacht Captain manage their expectations?

You cannot!

There is no way to perfectly match the expectations an Owner has built up over the years as he waits patiently for delivery day. To take it further; how should the Captain deliver bad news to this Yacht Owner? News that may reflect that the yacht does not function in the manner presented during the design, sales, purchase cycle. Or on charter, the promotional photo that shows all watersports in use, seemingly on demand, is a guest expectation that cannot be delivered; with the Captain trying to respect the legal obligation for hours of work and rest.

The honeymoon is now over and there is a risk of a breakdown in the Owner / Captain relationship, there is nobody else in the room. The photo of the ribbon cutting may be sitting in a frame or the charter brochure open on the web browser, but the actors belong to a time long-forgotten, all shortcomings are directed to the Captain.

This is a scene in which I have had a walk on role many times; as have most Captains. Standing before an unhappy Yacht Owner for an operational shortcoming that was built into the yacht with no way to address, is a humbling experience. And, as awkward as an operational ‘moment of truth’ can be, it is preferable to the personal rebuke that can’t be blamed on a technical or manning deficiency.

The Owner / Captain relationship has a sense of intimacy. The Yacht Owner spends significant time on the yacht, and the Captain is brought into their World. Many new Captains are swept up in the intoxication of being within this inner sanctum, wiser Captains maintain some separation knowing that such personal intimacy is fragile and can easily fracture – often without warning.

I recall from my own career, a time when I would greet the Principal on every arrival and departure from the yacht. This was and remains an accepted practice; one learnt from observing my former Captains, and absorbed automatically into my own Captaincy. It was some years into my time with a Yacht Owner when, during a particularly challenging conversation, he said, “and why do you meet me every time I move? Can’t I have some privacy?” It seemed such a small point but, over time, it had catalysed into a real annoyance for this exceptional Yacht Owner.

It should not have escalated to this level, but a Yacht Owner is not normally driven to address the issues of their day to their staff unless it is of a serious nature. Too often, a Yacht Captain’s success is measured only by the departing comment of the yacht owner who says, “Thank you, we had a great time.” This may be authentic or, likely the yacht owner is not ready to invest the time to deconstruct the trip at that point. Not unlike the automatic response when the ever-friendly waiter asks, “did you enjoy your meal?” The question is more rhetorical than a real enquiry of the dining experience.

I was only awakened to this cold reality when an Owner’s Representative confronted me with a concern of the Yacht Owner. I held my position that the Yacht Owner expressed gratitude and pleasure with the last visit. The Representative quickly cut me down and made clear what is said onboard is very different to the detailed debrief he received in the office some days later. It was a growth and career inflection point for me; I would no longer take for granted any Owner’s praise or make assumptions based on yachting’s normal practices.

I did not enjoy being admonished by the Owner’s Representative, but he caught what could have been a fatal rupture in an otherwise successful relationship. From that time on, we worked together to ensure the good health of the Owner / Captain relationship. I would no longer take for granted the warm smile on departure and would readily seek the ‘truth’ from my colleague in the Family Office.

OnlyCaptains seeks to support Captains and Yacht Owners through their model of Search, Select, Place and Mentor. We have lived the Owner / Captain relationship and realise it is the cornerstone of a successful yacht experience. We are the knowing advisor; able to listen, reflect and interpret the concerns of a Yacht Owner and use this insight to help coach the Captain to their success.

By |2021-06-14T09:41:02+02:00August 20th, 2020|captains, leadership, mentoring, yachtowners|

## How I Learnt What Was Missing From My Yacht Captaincy

Co-founder of OnlyCaptains Brendan O’Shannassy talks of his own experience of mentorship and how this experience awakened the need for this service in yachting.

In my early yachting career I often felt the ‘imposter’ as I stepped onto the bridge ready to manoeuvre. As my experience increased this feeling reduced but there was still something missing. There is a good chance that, like many of my peers, I may have even managed a career without knowing what the ‘secret sauce’ to achieving true technical competence in shiphandling was. This same observation could be extended to decision making, counselling and the many daily challenges of yacht command – I had the ticket, but did I have the knowledge?

I had received great support by yacht captains through my career but there had been no structured mentoring and learning plan in place. And, often it seemed, that the issue of a Master’s CoC was viewed as the end game, and not the beginning of the Captaincy journey.

Taking a break from yachting to train and work as a marine pilot, I entered a formal training / mentoring programme. New entry pilots are assigned a mentor, a more senior pilot that inducts, trains and supports them in their struggle to move from former captain and nervous shiphandler to competent pilot. A competent pilot who is expected to; step onboard any ship, at any hour, in any weather, take command and bring that ship safely to the port. Whilst my mentoring was focussed on ship handling, I found the learning extended far beyond the bridge; weaving the lessons into my wider Captaincy responsibilities also improved my performance, enjoyment and efficiency.

Returning to shiphandling, I finally understood the importance of defining and communicating the metrics of a manoeuvre. Everything fell into place. What speed? when and why? These need to be known at all times and this must be shared with all on the bridge. It is so important because at all times the entire bridge team must be able to assess every action of the manoeuvring Captain / Pilot against an earlier briefed and agreed plan. The Captain must then be able to communicate any deviation and why.

By communicating this it does not limit the Captain, it allows the Captain to vary the plan in response to the conditions and circumstances at the time. It also allows a challenge from other members of the bridge team if the deviation is not validated, – this is the philosophy of ‘challenge and response’ that is the bedrock of effective bridge teams.

During my first days of induction as a pilot I observed several senior pilots prior to being assigned my mentor. I was intrigued that the more experienced pilots exercised greater diligence with their pilotage briefings than those more junior. Their sketches of the ship’s planned manoeuvre into port were precise, their briefing books had photographs of landmarks relative to the pilotage and during execution they communicated to the entire bridge team what they were doing and why.

I had never seen this or, thought to do this myself, on yachts. Up to this point I would take control at some point on the approach and, whilst I would seek port information from the pilot, I really set my own approach speeds, headings and approach angles based on my best judgement at the time. Often I was monitoring my speed on the simple ‘ten-through-one’ method of checking speed reductions against the last ten cables. Although an effective approach, it is crude, not tailored to wind and tide, and a little too cautious with modern yachts. Through all the training and sea service to Yacht Captain, I had not learnt to communicate my intentions to the bridge team and I may as well have been there on my own – almost without saying, this extended to all leadership decisions.

After induction I was assigned a mentor pilot that was more than I could have hoped for. Ian had emigrated from the United Kingdom to Australia and with nineteen years as a Thames Pilot he had learnt from some of the best in the World and had experienced all the conditions that UK waters could offer – this made Ian a compelling mentor.

The North Queensland commercial port was a very different environment to my yachting experience. It had large tides, strong currents, steady wind and then strong gusts form the opposite direction. It was as unforgiving as I could imagine. The ships were also very different; replacing my delightfully over-powered and easy to manoeuvre yachts, were single screw bulk carriers that were also not as reliable as their Captains would have you believe.

All of this Ian took in his stride, in fact more than that, he actually expected everything to go wrong on every pilotage. He planned for it and whilst I would never be as bold as to call him out, I think he was at times disappointed when yet another of his ship moves went seamlessly. During one memorable departure of a fully loaded Cape Size (a Cape Size ship is 280 metres long, 52 metres wide and when loaded 18 metres deep) Ian turned to me and said, “see that?” I didn’t. My body language was enough to confirm this and asked I remind him after the departure.

Ian took the ship to the port limits safely, we both departed by helicopter and walking back from the helipad after the four-minute flight I asked him what I missed. He recalled that when he had asked the question, the bow of the ship was being pushed back by the water resistance as the ship ‘cut’ across the berth pocket.

This might need some explaining. The berth pocket was deeper than the departure channel and departing the berth needed the ship to move from 5.0m under keel clearance (UKC) to 0.9 metre (UKC). This required a wall of water 280m long and 18m high to be relocated through a very small gap – the water did not appreciate being forced through the gap and pushed against the ship.

In the departure we had shared, Ian had observed the ship being pushed back by this wall of water and the bow was moving 0.2 knots in the wrong direction, a speed almost imperceptible to the eye, however Ian was using both the pilot’s precise navigation unit and his highly-tuned sense from so many manoeuvres.

Ian increased the power on the ship astern to move the pivot point to his advantage and also the forward tug was increased to lifting off (pulling) at three quarter power to recover the bow. The entire event was observed, acted upon and rectified within two minutes. Ian’s point was, if it had not have been acted upon at that point it would have been very dangerous, and with a smirk he asked, “Do you know the fastest thing in the World?” I returned the smile and let the story play out…

“Brendan, the fastest thing in the World is a fully laden Cape Size bulk carrier moving half a knot in the wrong direction!”

It was a great lesson and, as the months progressed, and I moved from observer to the pilot executing the pilotage, the lessons still flowed. I never accepted anything Ian shared without chasing him with follow-up questions. He warmed to this and my shiphandling education accelerated at a rate where I began to surprise myself with my ability to anticipate and react to seemingly unlinked events. When I was a solo pilot and there was a complex move I would speak with Ian before boarding the ship to communicate my plan and build my confidence. My first call on completion would be to Ian to debrief; he was so good, he could picture the move from the call.

Ian had given me tools to use and these were centred around two aspects; the plan and the team available as a resource.

The planning began well before arriving to the ship. We would do the simple things of checking the radio battery was fully charged but, went further, and put a spare battery in our pockets ‘just in case’. The portable pilotage unit never failed, but nevertheless we turned it on and calibrated it ashore, every time. We would visit the control tower to look at their weather information; sure, we could look to reliable weather forecasting from our mobile phones but the control tower had real data from wind sensors on the docks and we could also look to an array of cameras that would also show detail as small as the wavelets on the water. The actual ship movement plan included; speed reference points, headings, abort points and of course final docking plan. This did not vary from the training I had received and, had perhaps previously belittled; it was just my mentor pilot had sewn it all together.

Ian had coached me to communicate each of these references in real time during the pilotage and the manoeuvring. If at 5 cables to the berth the plan was to be at 5 knots and the ship was at 6 knots, I would now say “the ship is above our agreed plan and I am comfortable with this but will reduce speed and report again as we pass 4 knots.” This narrative continued across all aspects of the plan and the ship’s Captain and Bridge team. Ian had trained me to make sure the dialogue was both ways, as the crew’s opinions were sought to the point of being demanded. It changed everything. I was no longer ‘alone’ on the bridge, everyone was working with me, as Ian commented:

“You have multiplied your safety by the number of people now engaged”.

Transferring this to yachts, it is possible to gain benefit from any crew member, even when numbers are thin. The discipline of verbalising ship movements to anyone creates confidence and accountability – If you can’t communicate what you are doing, are you really in charge?

I was appointed a great mentor with a structured training plan, but this might not always be the case. I knew that moving forward if there was no, assigned mentor, I would identify the person I wanted to learn from, approach them and let them know I wanted to learn, an important factor as the mentee’s desire and commitment to learn is as crucial as the mentor’s role in supporting their development.

I find many of these great ship handling lessons can also be applied to life; Ian, taught me to:-

1. test and verify equipment (or ideas),
2. develop a plan,
3. communicate the plan,
4. amend the plan in sympathy to the changing conditions, and
5. engage others to support and challenge the plan.

The same sense of ‘going it alone’ that I felt on the bridge before Ian’s tutelage may well have spoken to my previous leadership endeavours as a Yacht Captain. My ability to communicate and embrace the support of a team to safely bring a large ship into port gave me a new framework with which to lead a team on returning to Europe where a new Yacht Command beckoned. I would not say it was a silver bullet to success, but it certainly helped – and I still have not stopped learning.

Back on the yacht I missed my mentor and sought to replace this support and extend it to all aspects of my command. Unfortunately, there was no knowledgeable Captain to air my professional challenges in confidence and, although I worked well with management, none had held seagoing command, so they could not provide effective counsel.

Often, Captains, are just expected to ‘get on and do the job’ we have the ‘ticket’ after all, shouldn’t that be enough? However, the reality is that whatever the perceived experience level, we cannot know everything and, it is a dangerous Captain who thinks he does. We are learning all the time and regularly encounter situations never faced before, where the decision making could benefit from confidential counsel with someone who has ‘lived’ experience and can add value to the decision making process and personal development.

OnlyCaptains commitment to our mentoring role was created in response to my experience and that of my fellow Captains who, when I spoke of my journey, asked ‘”how do we access a mentor?” We provide an answer by making sure command is not a solo affair by providing our Captains with support and mentoring that can help them grow and develop their skills as Captains and leaders.

By |2021-06-14T09:41:40+02:00August 9th, 2020|captains, leadership, mentoring, ship handling|

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May 18, 2016 · This equation can be re-arranged as : h 2 = g h ( T 2 π) 2 − k 2. So if you plot values of h 2 against h ( T 2 π) 2 you will get a straight line y = a x + b with slope a = g and y intercept b = − k 2, from which you can find the moment of inertia about the CG : I = m k 2. where m is the mass of the lamina. So the procedure is :. "/> elearnsecurity exploit development exam curse of the wolf

# Mass of lamina formula

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Sep 25, 2014 · Find the coordinates of the center of mass of the lamina bounded by the curves and . Plot both functions on the same graph along with the point to see if your answer makes sense. Find the coordinates of the center of mass of the lamina bounded by the curves and .. The centre of mass for the original shape is then determined from the centre of mass for a set of particles of different masses attached to a light lamina. 2M 1M 3M Uniform lamina mass M Light lamina with particle mass M at.

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This calculus video tutorial explains how to find the x and y coordinates of the centroid or the center of mass of the region bounded by one or two equations. We can find the center of mass between the two objects when we have know the mass and distances between the two objects. Formula to calculate center of mass between the two objects is given by: where, X = Center of mass [m] m 1 = Mass of object 1 [kg] m 2 = Mass of object 2 [kg] x 1 = Distance of object 1 from fixed point [m]. Leaf lamina mass and area are closely correlated with the photosynthetic capacity and competitive ability of plants, whereas leaf age has been demonstrated to affect physiological processes such as photosynthesis. However, it remains unknown whether the lamina mass vs. area scaling relationship is influenced by leaf age, which is important for understanding plant adaptive strategies and, more.

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Center of mass of two point masses. Center of mass of two point masses is given by: rCOM= m 1+m 2m 1r1+m 2r2. Example: In a carbon monoxide molecule, the carbon and the oxygen atoms are separated by a distance 1.2×10 −10m. Then find the radius of carbon atom is (Assume both atoms touch each other). Solution:. The 𝑥 -position of the center of mass of the lamina is given by 𝑥 = = 5 5 7. The result can be confirmed using the following formula: 𝑥 = 𝑚 𝑥 + 𝑚 𝑥 𝑚 + 𝑚. Substituting known values gives 𝑥 = 𝑚 × 4 + ( 𝑚 × 1 3) 𝑚. The mass of the triangular lamina is the norm of the vector . Formula can be found in. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two. L => Length of square M = > Mass of Square Lamina 2) By Perpendicular axes Theorem and Symmetry, 3) By Parallel axes Theorem, Hence, Moment of Inertia about an axis touching it's side in the plane of Lamina is . Advertisement Advertisement Erik9422 Erik9422 Answer: I = ml²/6. Recorded on June 30, 2011 using a Flip Video camera.

6. 23. · Its density at a point (x, y) in the region R is ρ (x, y). The total mass of the lamina is expressed through the double integral as follows: The static moment of the lamina about the -axis is given by the formula. Similarly, the. The center of mass is used for studying the motion of the body. The center of gravity is used to understand the stability of the body. 6. The center of mass is dependent upon mass distribution. On the other hand, the center of gravity is dependent upon the acceleration caused due to gravity in an object. 7. Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D = {(x, y) | 0 ≤ x ≤ 1, −1 ≤ y ≤ 1}; ρ(x, y) = 7xy 2 I got my mass Sep 14, 2015 · An object of mass M M is in the. We can find the center of mass between the two objects when we have know the mass and distances between the two objects. Formula to calculate center of mass between the two objects is given by: where, X = Center of mass [m] m 1 = Mass of object 1 [kg] m 2 = Mass of object 2 [kg] x 1 = Distance of object 1 from fixed point [m].

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First and foremost, we need to find the centre of mass. Since the square is uniform, the centre of mass is at the centre of the lamina. Weight now provides both a clockwise and anticlockwise moment. Clockwise, we now have. 0. 2 5 × 7 0 0 sin  1 5 ° + 0. 5 T. 123 Points. you can the position of the centre of mass of the given trapezium. for this firstly you have to divide the trapezium in two parts one is square ABCE of side ‘a’ and triangle AED such that AE = ED = ‘a’ and AD = a^ (1/2). now centre of mass of square will be at a distance of ‘a/2’ from BC and DC whereas centre of mass of .... Find the mass and center of mass of the lamina for each density. R: triangle with vertices (0, 0), (0, a), (a, a) (a) ρ = k. m = (x, y) = (b) ρ = ky. m = (x, y) = (c) ρ = kx. m = (x, y) = I found the answer for "m" in part (a) which is ka^2 / 2, which is correct but cant find the answers to the rest of them using the same formula. Help ....

The center of mass is used for studying the motion of the body. The center of gravity is used to understand the stability of the body. 6. The center of mass is dependent upon mass distribution. On the other hand, the center of gravity is dependent upon the acceleration caused due to gravity in an object. 7. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional object with constant density. With a double integral we can handle two dimensions and variable density. where M is the total mass, M y is the moment around the y -axis, and. The 𝑥 -position of the center of mass of the lamina is given by 𝑥 = = 5 5 7. The result can be confirmed using the following formula: 𝑥 = 𝑚 𝑥 + 𝑚 𝑥 𝑚 + 𝑚. Substituting known values gives 𝑥 = 𝑚 × 4 + ( 𝑚 × 1 3) 𝑚. Glossary center of mass the point at which the total mass of the system could be concentrated without changing the moment centroid the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case. Jun 07, 2022 · Search: Center Of Mass Calculator Symbolab. A lower static marging will result in less The center of mass of the coin is at some coordinate (x,y,z) in a system where the center of the coin is the origin Formula: Rx = (ma*xa + mb*xb) / (ma+mb) Where, Rx = Center of Mass or Gravity com Center of mass is the point at which entire mass is distributed only at that one.

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The centre of mass of a uniform lamina whose shape is that of R, is denoted by G. Use integration to determine the coordinates of G. FM2-Q , (3 9,) 4 5 G 1 R O y y x= 6 2 x. Created by T. Madas Created by T. Madas Question 3 (**+) The figure above shows the finite region R bounded by the coordinate axes, the curve. .

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Leaf lamina mass and area are closely correlated with the photosynthetic capacity and competitive ability of plants, whereas leaf age has been demonstrated to affect physiological processes such as photosynthesis. However, it remains unknown whether the lamina mass vs. area scaling relationship is influenced by leaf age, which is important for understanding plant adaptive strategies and, more. Find step-by-step Calculus solutions and your answer to the following textbook question: Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. D is the triangular region enclosed by the lines x = 0, y = x, and 2x + y = 6; $$p(x, y) = x^2$$..

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5.6.1 Use double integrals to locate the center of mass of a two-dimensional object. 5.6.2 Use double integrals to find the moment of inertia of a two-dimensional object. 5.6.3 Use triple integrals to locate the center of mass of a three-dimensional object. We have already discussed a few applications of multiple integrals, such as finding. Taking the limit as n→ ∞ n → ∞ gives the exact mass of the lamina: m = lim n→∞ n ∑ i=1ρf (x∗ i)Δx = ρ∫ b a f (x)dx m = lim n → ∞ ∑ n i = 1 ρ f ( x i ∗) Δ x = ρ ∫ a b f ( x) d x Next, we calculate the moment of the lamina with respect to the x x -axis.

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Search: Center Of Mass Calculator Symbolab.Calculate your own natural potential or find out if someone is "natty or not" 0 APK (Lastest Version) A rotation is a transformation in which the pre-image figure rotates or spins to the location of the image figure Symbolab is an advanced math education tool A separable differential equation is a differential equation that can be put in the. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional object with constant density. With a double integral we can handle two dimensions and variable density. where M is the total mass, M y is the moment around the y -axis, and. Firstly, we will assume that the square plate consists of a mass (M) and sides of length (L). The surface area of the plate A = L X L = L2 Further, we will explain the mass per unit area as: Surface density, ρ = M A M A = [ M L 2] Applying integration; Iplate = 1 / 6 M / L2 Find Moment of Inertia of a Square Plate along an Axis. Let us consider a rectangular lamina of mass having length and breadth . The surface area of the lamina is . Hence, the mass per unit area of the lamina is . Let us consider a thin strip of width at a distance from the centre of gravity . The area of the strip is , The mass of the strip is given by,. If the density were a constant, finding the total mass of the lamina would be easy: we would just multiply the density by the area. When the density isn't constant, we need to integrate instead. The mass of a little box of area around the point is essentially. The mass of the triangular lamina is the norm of the vector . Formula can be found in. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable. The mass of a little box of area d A around the point ( x, y) is essentially ρ ( x, y) d A. For the total mass of the lamina, we add up the boxes and take a limit to get. M = ∬ D ρ ( x, y) d A. This integral can be done in rectangular coordinates, polar coordinates, or by whatever method you prefer. Example 1: Find the mass of a lamina with. The mass of the triangular lamina is the norm of the vector . Formula can be found in. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable. I was reading about moment of inertia on Wikipedia and thought it was weird that it had common values for shapes like tetrehedron and cuboids but not triangular prisms or triangular plates, so I tr.

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5.6.2 Use double integrals to find the moment of inertia of a two-dimensional object. 5.6.3 Use triple integrals to locate the center of mass of a three-dimensional object. We have already discussed a few applications of multiple integrals , such as finding areas, volumes, and the average value of a function over a bounded region.

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Suppose you have a lamina bounded by the curves , whose area was computed above and you want to compute the center of mass. If you are not sure about the relative positions of the two curves, it is a good idea to plot them both. UY1: Centre Of Mass Of A Cone. September 15, 2015 by Mini Physics. Find the centre of mass of an uniform cone of height h h and radius R R. Let the density of the cone be ρ ρ. It is obvious from the diagram that the x and y components of the centre of mass of a cone is 0: xCM = 0 yCM = 0 x C M = 0 y C M = 0. Hence, we just need to find zCM z C M. The centre of mass of a uniform lamina whose shape is that of R, is denoted by G. Use integration to determine the coordinates of G. FM2-Q , (3 9,) 4 5 G 1 R O y y x= 6 2 x. Created by T. Madas Created by T. Madas Question 3 (**+) The figure above shows the finite region R bounded by the coordinate axes, the curve. Search titles only.

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Prove, by integration that the centre of mass of the lamina is at a distance 2h/3 from X. Heres what I don't get: It states in my book the equations for centre of mass for a lamina: = divided by. and. = divided by. In the solutions for the question above, they've used a different equation to get , check in the attached picture below.. Verified by Toppr. The lamina (ΔLMN) may be subdivided into narrow strips each parallel to the base (MN) as shown in Fig., By symmetry each strip has its centre of mass at its midpoint. If we join the midpoint of all the strips we get the median LP. The centre of mass of the triangle as a whole therefore, has to lie on the median LP.. Recorded on June 30, 2011 using a Flip Video camera. Find Mx, My, and for the laminas of uniform density ρ bounded by the graphs of the equations. 0 votes Find Mx, My, and for the laminas of uniform density ρ bounded by the graphs of the equations. y = x2 , y = x3 moments centroids asked Feb 17, 2015 in CALCULUS by anonymous 2 Answers 0 votes Step 1: The equations of the curves are and. Find the centre of mass of a triangular lamina. Medium Solution Verified by Toppr The lamina (ΔLMN) may be subdivided into narrow strips each parallel to the base (MN) as shown in Fig., By symmetry each strip has its centre of mass at its midpoint. If we join the midpoint of all the strips we get the median LP. Above equation will be termed as the first moment of mass about the OY axis and it is used to determine the centroid or centre of gravity of the mass of the lamina. If the first moment of mass will again multiplied with the vertical distance between the C.G of the mass and axis OY, then we will have second moment of mass i.e. M.X 2.

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Feb 27, 2019 · It is measured by the mass of the body. The moment of inertia (I) of a body is a measure of its ability to resist change in its rotational state of motion. Consider a triangular lamina of base (b), altitude (h) and mass (M). Surface mass density is mass per unit area of the lamina. \ (\sigma =\frac {M} {\left ( \frac {1} {2}bh \right)}=\frac .... Find the distance of the centre of mass of the lamina from . AE. 25cm. 200g. 100g. 200g. 25cm. r. r. h. 2.*A uniform solid, S, is placed on a horizontal table. The solid is made from a right circular Dec 30, 2021 · The centre of. Taking the limit as n→ ∞ n → ∞ gives the exact mass of the lamina: m = lim n→∞ n ∑ i=1ρf (x∗ i)Δx = ρ∫ b a f (x)dx m = lim n → ∞ ∑ n i = 1 ρ f ( x i ∗) Δ x = ρ ∫ a b f ( x) d x Next, we calculate the moment of the lamina with respect to the x x -axis. The mass of the triangular lamina is the norm of the vector . Formula can be found in. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two. Sep 25, 2014 · Find the coordinates of the center of mass of the lamina bounded by the curves and . Plot both functions on the same graph along with the point to see if your answer makes sense. Find the coordinates of the center of mass of the lamina bounded by the curves and ..

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(a) Find the mass of the lamina. Find the center of mass of a lamina that occupies the region bounded by the parabola x = 1 - y^2 and the coordinate axes in the first quadrant if the lamina has. If the density were a constant, finding the total mass of the lamina would be easy: we would just multiply the density by the area. When the density isn't constant, we need to integrate instead. The mass of a little box of area around the point is essentially. Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D = {(x, y) | 0 ≤ x ≤ 1, −1 ≤ y ≤ 1}; ρ(x, y) = 7xy 2 I got my mass to be 7/3 but I'm not sure how to go about finding the center of mass. The mass of the triangular lamina is the norm of the vector . Formula can be found in. 15.3 Moment and Center of Mass. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two.

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6. 23. · Its density at a point (x, y) in the region R is ρ (x, y). The total mass of the lamina is expressed through the double integral as follows: The static moment of the lamina about the -axis is given by the formula. Similarly, the static moment of the lamina about the -axis is. The coordinates of the center of mass of a lamina. Let us consider a rectangular lamina of mass having length and breadth . The surface area of the lamina is . Hence, the mass per unit area of the lamina is . Let us consider a thin strip of width at a distance from the centre of gravity . The area of the strip is , The mass of the strip is given by,. Find step-by-step Calculus solutions and your answer to the following textbook question: Find the mass and center of mass of the lamina that occupies the region D and has the given density D is the triangular region enclosed by the lines x = 0, y = x, and 2x + y = 6; $$p(x, y) = x^2$$. Basically, a planar lamina is defined as a figure (a closed set) D of a finite area in a plane, with some mass m. This is useful in calculating moments of inertia or center of mass for a constant density, because the mass of a lamina is proportional to its area. In a case of a variable density, given by some (non-negative) surface density function. Find the mass of the lamina that occupies the region D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with the density function ρ(x, y) = xye^(x+y) - 12890381 andreacisneros9853 andreacisneros9853 06/29/2019 Mathematics College answered 1. If the point is omitted, you get a moment of inertia matrix around the center of mass : This is equivalent to specifying the center point by RegionCentroid : Compute the moment of. Now the next step involves integration where we integrate over the entire lamina. Mechanical Engineering Problems With Solution - DocShare.tips Radius of gyration. A uniform disc of mass 100 g has a diameter of 10 cm. Centre of gravity (qualitative only) with examples of some regular bodies and irregular lamina.

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May 03, 2012 · 1) A triangle with sides 3", 4", and 5" respectively. So this is a right triangle. Set up a coordinate system so the right angle is at the origin, one vertex is to (3, 0), and another at (0, 4). The equation of the hypotenuse is y= - (4/3)x+ 4. The area of the triangle is, of course (1/2) (3) (4)= 6 so the x-coordinate of the centroid is and .... Then the mass m of the lamina, and the coordinates (x, y) of the center of mass of the lamina, are given by the following formulas: m = , x = , y = (the x and y are there, in part, because of the teeter-totter discussion above) Theorem 20.1: Given any density function (x, y) defined on a domain D, the mass m of the lamina, and the. 6. 23. · Its density at a point (x, y) in the region R is ρ (x, y). The total mass of the lamina is expressed through the double integral as follows: The static moment of the lamina about the -axis is given by the formula. Similarly, the static moment of the lamina about the -axis is. The coordinates of the center of mass of a lamina. Professor Mahesh Wagh has pledged to eradicate the fear of Mathematics from all those students who are afraid of studying this subject. His experience of teaching mathematics stretches over a time span of around 13 years. He has earned a degree in computer engineering from Mumbai University. Apart from this, he also has industrial experience of. A plane lamina occupies the region in the xy-plane between the two circles x2 + y2 = 4 and x2 + y2 = 49 and above the x-axis.The center of . mass of the lamina if its mass density is is: . The center of the mass of the lamina can be computed by using the formula:. The mass of the region R can be estimated by using the formula:. To determine the area by using polar coordinates, we have:.

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Nutritional support to strengthen the connective tissue of the hooves should be provided by giving Farrier's Formula ® original or Farrier's Formula ® Double Strength along with Life Data ® Lamina Formula. Adult Horses. For each 1,000 lbs. (450 kg) of body weight feed one standard kitchen measuring cup (237 ml or 170 g) of product per day. Jan 11, 2020 · The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass 4kg. asked Jan 25, 2020 in Physics by Nakul01 ( 37.0k points) jee main 2020. Prove, by integration that the centre of mass of the lamina is at a distance 2h/3 from X. Heres what I don't get: It states in my book the equations for centre of mass for a lamina: = divided by. and. =. The green region is the area to compute the center of mass. First, we must find the mass of the lamina In our problem, we have that: f (x) = √ 4 9 − x , g (x) = 0, a = 0, b = 4 9 . As the lamina is homogeneous then the density is m. The formula given here is for the center of mass in one dimension We can see from the Figure that the line $$y = 3x + 2$$ lies above the cubic parabola $$y = {x^3}$$ on the interval $$\left[ { – 1,2} \right] Not so! We all have different. If the point is omitted, you get a moment of inertia matrix around the center of mass : This is equivalent to specifying the center point by RegionCentroid : Compute the moment of. The formula given here is for the center of mass in one dimension We can see from the Figure that the line \(y = 3x + 2$$ lies above the cubic parabola $$y = {x^3}$$ on the interval $$\left[ { – 1,2} \right] Not so! We all have different. Search: Center Of Mass Calculator Symbolab.Calculate your own natural potential or find out if someone is "natty or not" 0 APK (Lastest Version) A rotation is a transformation in which the pre-image figure rotates or spins to the location of the image figure Symbolab is an advanced math education tool A separable differential equation is a differential equation that can be put in the. This video explains the formula to compute the mass of lamina based on Howard Anton's Calculus Text, and then show an example using iterated integral. This video explains the formula to compute. The center of mass is a position defined relative to an object or system of objects. It is the average position of all the parts of the system, weighted according to their masses. For simple rigid objects with uniform density, the center of mass is located at the centroid. For example, the center of mass of a uniform disc shape would be at its. Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D = {(x, y) | 0 ≤ x ≤ 1, −1 ≤ y ≤ 1}; ρ(x, y) = 7xy 2 I got my mass to be 7/3 but I'm not sure how to go about finding the center of mass. and Static Moments of a Lamina. Suppose we have a lamina which occupies a region \(R$$ in the $$xy$$-plane and is made of non-homogeneous material. Its density at a point \(\left(.

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The mass of the lamina is m = doubleintegral_D rho(x, y) dA = integral^1_0 integral^2-2x_0 (3 + 7x + 2y) dy dx ; Question: Find the mass and center of a triangular lamina with vertices (0, 0), (1, 0), and (0, 2) if the density function is rho(x, y) = 3 + 7x + 2y. The triangle is shown in the figure (Note that the equation of the upper boundary ...
Nutritional support to strengthen the connective tissue of the hooves should be provided by giving Farrier's Formula ® original or Farrier's Formula ® Double Strength along with Life Data ® Lamina Formula. Adult Horses. For each 1,000 lbs. (450 kg) of body weight feed one standard kitchen measuring cup (237 ml or 170 g) of product per day.
6. 23. · Its density at a point (x, y) in the region R is ρ (x, y). The total mass of the lamina is expressed through the double integral as follows: The static moment of the lamina about the -axis is given by the formula. Similarly, the
To find the distance of the centre of mass from PQ we use the formula: Q P S R 4kg 2kg 6kg 5kg 5cm 12cm ... Lamina Plate Mass 2.25∏ 2.25∏ 120 120-4.5∏ x co-ord 3 12 6 X Y co-ord 6 6 5 Y Plate = Rectangular Lamina - Circle 1 - Circle 2 mx mX
We can find the center of mass between the two objects when we have know the mass and distances between the two objects. Formula to calculate center of mass between the two objects is given by: where, X = Center of mass [m] m 1 = Mass of object 1 [kg] m 2 = Mass of object 2 [kg] x 1 = Distance of object 1 from fixed point [m]. ...