**Problem Statement:**

A homogeneous pulley consists of two wheels that rotate together as one around the same axis. The moment of inertia of the pulley is I_{CM} = 40 kg m^{2}. The radii of the two wheels are respectively R_{1} = 1.2 m and R_{2} = 0.4 m. The masses that are attached to both sides of the pulley are m_{1} = 36 kg and m_{2} = 12 kg respectively (see figure). The initial height of the mass m_{1} is h_{1} = 5 m.

- Calculate the height at which the mass m
_{2}will rise. - Calculate the angular velocity of the pulley.

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**Solution:**

To determine the height at which the mass m_{2} will rise we must analyze what is the arc swept by a point of the periphery for each of the two wheels when the mass m_{1} goes down. In the following figure we have represented the initial (A) and final (B) states of the system constituted by the two masses and the pulley. We have also represented a point P_{1} of the periphery of the wheel of radius R_{1} and a point P_{2} of the periphery of the wheel of radius R_{2}.

Since the two wheels constituting the pulley rotate together as one, the angle φ swept by the points P_{1} and P_{2} will be the same in the same time interval. In addition, these points describe a circular movement, so we can write:

When m_{1} reaches the ground, the height h_{1} that it has descended will be equal to the value of the arc traveled by the point P_{1}. And the height h_{2} that the mass 2 has ascended will be equal to the value of the arc traveled by the point P_{2}. So:

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To calculate the angular velocity of the pulley and the speed of the masses we will use the principle of conservation of energy.

In the previous figure you can observe the initial (A) and final (B) states of the system as well as the origin of heights that we will use to calculate the gravitational energies. The mechanical energy in state A is the gravitational energy of mass 1 because the three bodies that make up the system are at rest:

In state B, the system will have gravitational energy because the mass 2 has ascended to the height h_{2}. It will also have the translational kinetic energy of the two masses and the rotational energy of the pulley:

As no non-conservative force (friction) acts on the system, the mechanical energy is conserved:

We can also relate the linear velocity of each mass with the angular velocity of the pulley:

And substituting in the energy conservation equation we get:

The only unknown variable of the above equation is the angular velocity of the pulley. After solving and substituting the givens as well as the height h_{2} previously calculated we get:

In the problem we have used g = 10 m/s^{2}.

Do not forget to include the units in the results.

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