Calculating Kinetic Energy with Rotational Motion

[embudini]

Homework Statement

A student sits at rest on a piano stool that can rotate without friction. The moment of inertia of the student-stool system is 3.6 kg·m2. A second student tosses a 1.7 kg mass with a speed of 3.0 m/s to the student on the stool, who catches it at a distance of 0.38 m from the axis of rotation. The final angular speed is 0.504 rad/s.
(a) Does the kinetic energy of the mass-student-stool system increase, decrease, or stay the same as the mass is caught?

(b) Calculate the initial and final kinetic energy of the system.

Homework Equations

W=Kf-Ki
ω=v/r

The Attempt at a Solution

All I've gotten so far is Ki=3.6+.5*1.7*3^2=11.25 J. I'm really stuck, please help!

 

[cepheid]

Welcome to PF embudini!

How does the moment of inertia of the system change after the person on the stool catches the mass?

How does the rotational kinetic energy of a system depend on its moment of inertia and on its angular velocity ω?

 

[embudini]

The moment of inertia would increase because the system has a greater mass.

The kinetic energy is .5mv^2+.5Iω^2, the combination of its linear and rotational kinetic energies. But I don't know what to do this these equations in this scenario. I have a lot of problems similar in concept to this one, but I just don't understand how to solve for them!

 

[cepheid]

The moment of inertia would increase because the system has a greater mass.

Yes, but how, specifically does it change? Hint: ignoring the mass of the person's arm, you can think of the mass that gets caught as being a point mass that is moving in a circle around the centre of rotation whose radius is the arm length. What is the moment of inertia of such a system? How does it combine with the moment of inertia of the person + stool?

The kinetic energy is .5mv^2+.5Iω^2, the combination of its linear and rotational kinetic energies. But I don't know what to do this these equations in this scenario. I have a lot of problems similar in concept to this one, but I just don't understand how to solve for them!

Yes, but after the catch happens, is there linear motion any longer?

 

[rwisz]

To calculate the kinetic energy of the system, we can use the equation K = 1/2Iω^2, where I is the moment of inertia and ω is the angular velocity.

(a) In this scenario, the kinetic energy of the system will increase as the mass is caught. This is because the mass adds to the moment of inertia of the system, and the final angular velocity is greater than the initial angular velocity.

(b) To calculate the initial kinetic energy, we can use the given information to find the initial angular velocity, using the equation ω = v/r. Plugging in the values, we get ω = 3.0/0.38 = 7.89 rad/s. Therefore, the initial kinetic energy is K = 1/2(3.6)(7.89^2) = 140.6 J.

To calculate the final kinetic energy, we can use the given final angular velocity of 0.504 rad/s. Plugging this into the equation, we get K = 1/2(3.6)(0.504^2) = 0.455 J.

Therefore, the initial kinetic energy is 140.6 J and the final kinetic energy is 0.455 J. We can see that the kinetic energy has decreased as the mass was caught, because some of the initial kinetic energy was converted into rotational kinetic energy.