# Tensions And Pulleys With Masses

The problem I am working on is:

"A block of mass m1 = 1.80 kg and a block of mass m2 = 6.30 kg are connected by a massless string over a pulley in the shape of a solid disk having radius R = 0.250 m and mass M = 10.0 kg. The fixed, wedge-shaped ramp makes an angle of θ = 30.0° as shown in the figure. The coefficient of kinetic friction is 0.360 for both blocks."

The provided diagram:

Determine the acceleration of the two blocks. (Enter the magnitude of the acceleration.)

Determine the tensions in the string on both sides of the pulley.

What I was wondering is why there are two different tension forces acting on the pulley? Could someone give me a descriptive answer? Also, does the mass of the pulley somehow affect the tension forces? Why exactly?

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## 2 Answers

Tension is a vector, so it has different directions on either side. For second question, imagine what would happen to tension if you had a pulley with the mass of the moon.

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Great point. To understand questions like this it is always a good idea to think of extreme situations. – ja72 Dec 1 '12 at 14:37
Oh, yes...The pulley would definitely have more rotational inertia, and that would diminish the amount of force transmitted, right? – Mack Dec 1 '12 at 15:11
I've sort of run into a road-block trying to solve this problem. I try to solve for the acceleration, but I have no way of eliminating the unknowns from my equations; what still remains is the two tensions forces. So, I tried to figure out how they were related through torque on the pulley; and despite the fact that there is a relationship, it introduces another unknown, namely, the torque. How do I solve for the acceleration? – Mack Dec 1 '12 at 15:19
I actually figured it out. However, I have a new question: if the tension forces on either side of the pulley are different, then why is there only one acceleration for the two blocks and the particles on the outer rim of the pulley? – Mack Dec 1 '12 at 15:43
There is two accelerations, of the same magnitude, acceleration is a vector too. A pulley is an idealized solid contraption, where motion is restriced to circular. – Hobo Dec 1 '12 at 22:06

Using the principle of virtual work, if you move the blocks a distance a, the inclined block is lowered by an amount equal to $a\sin(\theta)$, meaning that it gains energy $m_2 ga\sin(\theta)$. The total moving mass is $m_1 + m_2$, so that the acceleration is the same as for a mass $m_1 + m_2$ in 1 dimension with a force $m_2 g \sin(\theta)$, so that

$$a = {m_2 \over m_1 + m_2} g \sin\theta$$

This is how you solve these types of problems, it's equivalent to writing the Lagrangian, but more elementary sounding.

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