For the system of two crates and a pulley in Figure P9.14, (m1 = 28.5 kg, m2 = 10 kg, mpulley = 8 kg), what fraction of the total kinetic energy resides in the pulley?

To determine the fraction of the total kinetic energy that resides in the pulley, we need to first calculate the total kinetic energy of the system.

The kinetic energy of an object is given by the formula:

KE = 1/2 * m * v^2

where KE is the kinetic energy, m is the mass, and v is the velocity of the object.

In this system, we have two crates and a pulley. Let's denote the kinetic energy of the first crate as KE1, the kinetic energy of the second crate as KE2, and the kinetic energy of the pulley as KEpulley.

The total kinetic energy of the system is the sum of the kinetic energies of the individual components:

Total KE = KE1 + KE2 + KEpulley

To calculate KE1, we need to find the velocity of the first crate. Since the crates are connected by a rope passing over a pulley, their velocities are related. Let's assume the magnitude of the acceleration of the system is a. The velocity of the first crate can be calculated using the formula:

v1 = a * t

where t is the time taken for the crate to travel a certain distance.

Similarly, the velocity of the second crate can be calculated as:

v2 = -a * t

since the motion of the two crates is in opposite directions.

To calculate the velocities, we need to find the acceleration of the system, which can be calculated using Newton's second law:

Sum of forces = mass * acceleration

In this case, the only forces acting on the system are the gravitational forces on the crates and the tension in the rope. The tension in the rope is equal to the force needed to accelerate the system, so:

Tension = (m1 + m2 + mpulley) * a

From this equation, we can calculate the acceleration, a.

Once we have the velocities, we can calculate the kinetic energies of the crates using the formula mentioned earlier.

The kinetic energy of the pulley can be calculated by considering that the pulley does not translate but only rotates, and its kinetic energy is given by:

KEpulley = 1/2 * I * ω^2

where I is the moment of inertia of the pulley and ω is its angular velocity.

The moment of inertia of a solid disc is given by:

I = (1/2) * m * r^2

where m is the mass of the pulley and r is its radius.

The angular velocity, ω, can be calculated using the relation between linear velocity and angular velocity:

v = r * ω

From this equation, we can solve for ω.

Once we have the kinetic energy of each component, we can calculate the fraction of the total kinetic energy residing in the pulley by dividing the kinetic energy of the pulley by the total kinetic energy:

Fraction = KEpulley / (KE1 + KE2 + KEpulley)

By plugging in the given values for the masses (m1 = 28.5 kg, m2 = 10 kg, mpulley = 8 kg) and following the steps outlined, you can determine the fraction of the total kinetic energy that resides in the pulley.