The two masses (m1 = 5.0 kg and m2 = 3.0 kg) in the Atwood's machine shown in Figure 10-23 are released from rest, with m1 at a height of 0.87 m above the floor. When m1 hits the ground its speed is 1.4 m/s. Assume that the pulley is a uniform disk with a radius of 12 cm. Determine the pulley's mass.

To determine the pulley's mass, we need to use the principles of conservation of energy and Newton's laws of motion. Here's how to approach the problem:

1. Start by calculating the gravitational potential energy of mass m1 when it is at a height of 0.87 m. The formula for gravitational potential energy is given as U = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height. So in this case, U1 = m1 * g * h1.

2. The energy from m1 is transferred to m2, causing it to accelerate downward. Let's call the final velocity of m2 as v2f.

3. The final velocity of m1 can be found using the equation v^2 = u^2 + 2as, where u is the initial velocity (0 m/s), a is the acceleration, and s is the distance (height) traveled. Rearranging the equation, we get u1^2 = v1f^2 - 2as -> 0 = (1.4 m/s)^2 - 2*(-9.8 m/s^2)*0.87 m. Solve for v1f.

4. Next, we'll consider the pulley. The pulley is a uniform disk, which means that its moment of inertia (I) is given by I = (1/2) * m * r^2, where m is the mass of the pulley and r is the radius.

5. The tension in the string connecting m1 and m2 causes a torque on the pulley, causing it to rotate. The torque can be given as T = I * alpha, where T is the tension in the string and alpha is the angular acceleration of the pulley.

6. Using Newton's second law of motion, the acceleration of m2 can be expressed as F = m2 * a, where F is the net force acting on m2. The tension in the string provides the net force, so T = m2 * a.

7. Since the tension in the string is the same throughout, we can equate T to the torque expression: T = I * alpha. Rearrange the equation to solve for alpha.

8. The angular acceleration of the pulley can be related to the linear acceleration of m2 by alpha = a / r, where r is the radius of the pulley.

9. Substitute the expression for alpha into the equation from step 7, and solve for m2.

10. Substitute the value of m2 into the equation from step 6, and solve for T.

11. The tension (T) can also be expressed in terms of m1 and g as T = m1 * g - m2 * g. Rearrange the equation to solve for m1.

12. Finally, substitute the values of m1, m2, and g, and solve for the mass of the pulley (m).

Follow these steps, perform the necessary calculations, and you will be able to determine the mass of the pulley in the Atwood's machine.

You know the energy gained (mgh)

You know the final KE of all masses, so the rest of the KE must be the pulley.

KE=I*w

I assume you will model the pulley as a disk of some radius r. w=vfinal/r

Solve for masspulley.