Figure 12-2. The book is in equilibrium; the net force on it is zero.
Solution: The forces at the point where the three cords join must add to zero. The vertical component of F A is the weight; the horizontal components of F A and F B cancel. Therefore, F A = 2260 N and F B = 1130 N.
Figure 12-4. Although the net force on it is zero, the ruler will move (rotate). A pair of equal forces acting in opposite directions but at different points on an object (as shown here) is referred to as a couple .
Figure 12-5. A lever can “multiply” your force. Solution: One way is to lengthen the lever, perhaps with a pipe slid over the end as shown. Another way is to move the small rock closer to the big rock; this increases the ratio of the two lever arms considerably.
Figure 12-6. (a) Two children on a seesaw. (b) Free-body diagram of the board. Solution: Choose to take the axis of rotation to be the pivot point. We have two equations, the force equation and the torque equation. The force equation does not give us x, but the torque equation does; x = 3.0 m.
Figure 12-7. A cantilever. The force vectors shown are hypothetical—one may even have a different direction.
Solution: a. Calculate torques around the point where F J acts; this gives F M = 400 N. b. This situation gives the same torque equation, so F M = 400 N still.
Solution: We have three unknowns, F T and the two components of F H . We know the angle of the cable, though, so we really need find only one component of F H . Taking the torque around the point where F T and mg act, and using the force equations, we find F Hy = 123 N, F Hx = 687 N, and F T = 793 N.
Solution: We have three unknowns, the force exerted by the wall and the two forces exerted by the floor (the normal force and the static frictional force). The normal force equals the weight, 118 N (from the vertical force equation), and the static frictional force equals the force exerted by the wall; writing the torque equation around the point where the ladder touches the floor gives F W = 44 N.
13 static equilibrium and elasticity
Static Equilibrium and Elasticity Topic 12
Lecture Outline 12.1 Static 12.2 The Condition For Equilibrium 12.3 Static Problem
12.1 Static <ul><li>Net force and net torque on an object/objects are both zero </li></ul><ul><li>Forces and torques acting on and within a structure are in equilibrium </li></ul>
12.2 The Conditions for Equilibrium <ul><li>Object at rest, net force is zero </li></ul><ul><li>Example: Book on a table, two forces acting on it </li></ul><ul><ul><li>Gravitational force </li></ul></ul><ul><ul><li>Normal force </li></ul></ul><ul><li>Book at rest, acceleration zero, total force is zero </li></ul><ul><li>Gravitational force and normal force balance each other </li></ul><ul><li>The book is in equilibrium (“balance” in Latin) </li></ul>
<ul><li>The First Condition For Equilibrium: </li></ul>An object with forces acting on it, but with zero net force, is said to be in equilibrium.
Example 12-1: Chandelier cord tension. Calculate the tensions A and B in the two cords that are connected to the vertical cord supporting the 200-kg chandelier shown. Ignore the mass of the cords.
Consider an object exerted by two forces with the same magnitude, pointing at the opposite direction The total force add up to be zero, but the position of the force might rotate the object For an object to be at rest, the total torque also must be zero. The second condition of equilibrium:
Conceptual Example 12-2: A lever. This bar is being used as a lever to pry up a large rock. The small rock acts as a fulcrum (pivot point). The force required at the long end of the bar can be quite a bit smaller than the rock’s weight mg , since it is the torques that balance in the rotation about the fulcrum. If, however, the leverage isn’t sufficient, and the large rock isn’t budged, what are two ways to increase the leverage?
12.3 Solving Statics Problems Example 12-3: Balancing a seesaw. A board of mass M = 2.0 kg serves as a seesaw for two children. Child A has a mass of 30 kg and sits 2.5 m from the pivot point, P (his center of gravity is 2.5 m from the pivot). At what distance x from the pivot must child B, of mass 25 kg, place herself to balance the seesaw? Assume the board is uniform and centered over the pivot.
If a force in your solution comes out negative (as A will here), it just means that it’s in the opposite direction from the one you chose. This is trivial to fix, so don’t worry about getting all the signs of the forces right before you start solving.
Example 12-4: Force exerted by biceps muscle. How much force must the biceps muscle exert when a 5.0-kg ball is held in the hand (a) with the arm horizontal, and (b) when the arm is at a 45° angle? The biceps muscle is connected to the forearm by a tendon attached 5.0 cm from the elbow joint. Assume that the mass of forearm and hand together is 2.0 kg and their CG is as shown.
Example 12-5: Hinged beam and cable. A uniform beam, 2.20 m long with mass m = 25.0 kg, is mounted by a small hinge on a wall. The beam is held in a horizontal position by a cable that makes an angle θ = 30.0 °. The beam supports a sign of mass M = 28.0 kg suspended from its end. Determine the components of the force H that the (smooth) hinge exerts on the beam, and the tension F T in the supporting cable.
Example 12-6: Ladder. A 5.0-m-long ladder leans against a smooth wall at a point 4.0 m above a cement floor. The ladder is uniform and has mass m = 12.0 kg. Assuming the wall is frictionless (but the floor is not), determine the forces exerted on the ladder by the floor and by the wall.