Rotational energy - Two pulleys of different radii

Problem Statement:

A disk of mass M 1 = 6 kg and radius R1 = 0.8 m can rotate around a horizontal axis, as indicated in the figure. A rope without mass is wrapped around the periphery of the disk, passing through another disk of mass M 2 = 2 kg and radius R2 = 0.5 m and is tied at the end to a block of mass m = 3 kg. There is no friction on the axes and the rope does not slip.

Assuming that the system is initially at rest when released, determine the speed of the block when descending a height h = 2 m.

Givens: The moment of inertia of a disc with respect to an axis that passes through its center of mass is: ICM = (1/2)MR2

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Solution:

To determine the speed of the block we are going to use the principle of conservation of energy. The following figure shows the initial (A) and final (B) states of the system constituted by the two disks and the block. The figure includes the origin of heights that we will use as a reference to calculate the potential energy.

In state A the energy of the system is the gravitational potential energy of the block. The discs also have potential energy, but it will not vary between state A and B and therefore we do not take it into account. Therefore, the energy in state A is:

In state B the mass m has lost all its potential energy but it has kinetic energy. On the other hand, the discs are rotating, so they will have rotational energy. The total energy in state B is given by:

The total energy of the system is conserved, since no non-conservative force acts on it:

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On the other hand, we can relate the angular velocity of each disk with the linear velocity of the mass m. As the rope does not slide, a point at the periphery of each disk will have the same linear velocity as the mass m. So:

After deducing the angular velocities we can substitute them in the energy conservation equation. We can also substitute the moments of inertia:

We can now deduce v and after substituting the givens we get:

In the problem we have used g = 10 m/s2.

Do not forget to include the units in the results of the problems.

 

 

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