For the system of two crates and a pulley in the figure below, what fraction of the total kinetic energy resides in the pulley? (Assume m1 = 14 kg, m2 = 10 kg and mpulley = 5 kg.)

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sorry, my diagram is not showing properly ..

it is a pulley with mass one hanging from the left side and mass two hanging from the right side

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

The kinetic energy of an object is given by the formula: KE = 1/2 * mass * velocity^2.

Since both crates are connected by the pulley, their velocities will be the same. Let's denote the velocity of both crates as v.

The total kinetic energy of the system can be calculated as the sum of the kinetic energies of each object:

KE_total = KE_m1 + KE_m2 + KE_pulley

To calculate KE_m1 (kinetic energy of crate m1), we need to know its mass and velocity. The mass of m1 is given as 14 kg. Since m1 is connected to the pulley, its velocity is the same as the pulling velocity, v.

KE_m1 = 1/2 * mass_m1 * velocity^2 = 1/2 * 14 kg * v^2 = 7v^2 kg*m^2/s^2

Similarly, to calculate KE_m2 (kinetic energy of crate m2), we need to know its mass and velocity. The mass of m2 is given as 10 kg. Again, since m2 is connected to the pulley, its velocity is the same as the pulling velocity, v.

KE_m2 = 1/2 * mass_m2 * velocity^2 = 1/2 * 10 kg * v^2 = 5v^2 kg*m^2/s^2

Now, let's calculate KE_pulley (kinetic energy of the pulley). The pulley is rotating, and its kinetic energy depends on both its mass and angular velocity. The equation for rotational kinetic energy is:

KE_pulley = 1/2 * moment_of_inertia * angular_velocity^2

The moment of inertia for a solid disk is given by the formula: moment_of_inertia = 1/2 * mass_pulley * radius^2.

In this case, the radius of the pulley is not given, so we cannot proceed with an exact calculation. However, we are asked to find the fraction of the total kinetic energy residing in the pulley, so we can ignore the exact value and focus on the relative fractions.

Since the radius of the pulley is a positive value, the moment of inertia and the kinetic energy of the pulley will be positive. Therefore, the fraction of the total kinetic energy residing in the pulley will be at least 0.

To find the maximum value of the fraction, we can assume that the pulley's kinetic energy is equal to the total kinetic energy:

KE_pulley = KE_total

Substituting the expressions we obtained earlier:

1/2 * moment_of_inertia * angular_velocity^2 = 7v^2 kg*m^2/s^2 + 5v^2 kg*m^2/s^2 + KE_pulley

Simplifying the equation:

moment_of_inertia * angular_velocity^2 = 12v^2 kg*m^2/s^2 + 2KE_pulley

Since both the moment of inertia and angular velocity are positive, the value of KE_pulley must be nonnegative, which means it cannot exceed 12v^2 kg*m^2/s^2.

Therefore, the fraction of the total kinetic energy residing in the pulley is between 0 and 12v^2/(7v^2 + 5v^2) = 12/12 = 1/6, or 16.67%.