A student sits at rest on a piano stool that can rotate without friction. The moment of inertia of the student-stool system is 4.6 kg m^2. A second student tosses a 1.5 kg mass with a speed of 2.9 m/s to the student on the stool, who catches it at a distance of 0.40 s from the axis of rotation. what is the initial and final kinetic engergy

To determine the initial and final kinetic energy in this scenario, we need to consider the conservation of angular momentum.

Angular momentum is given by the equation:

L = Iω

Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Initially, the student-stool system is at rest, so the initial angular momentum is zero.

If the second student tosses a mass, it will cause the student on the stool to start rotating. The moment of inertia of the student-stool system is given as 4.6 kg m^2.

The second student tosses a 1.5 kg mass with a speed of 2.9 m/s to the student on the stool. The student catches it at a distance of 0.40 m from the axis of rotation.

To calculate the final angular momentum, we can use the formula:

L = mvR

Where m is the mass, v is the velocity, and R is the distance from the axis of rotation.

Given that the mass of the tossed object is 1.5 kg, the velocity is 2.9 m/s, and the distance is 0.40 m, we can calculate the final angular momentum.

L = (1.5 kg)(2.9 m/s)(0.40 m)
L = 1.74 kg m^2/s

Since angular momentum is conserved, the initial and final angular momentum will be the same.

Initial angular momentum = Final angular momentum
0 = 1.74 kg m^2/s

The initial and final kinetic energy can be calculated from the equation:

KE = 0.5Iω^2

Since the initial angular velocity is zero, the initial kinetic energy is also zero.

KE_i = 0

For the final kinetic energy, we need to solve for the final angular velocity ω using the equation:

L = Iω

Since L = 1.74 kg m^2/s and I = 4.6 kg m^2, we can find ω.

1.74 kg m^2/s = (4.6 kg m^2)ω

ω = 1.74 kg m^2/s / 4.6 kg m^2
ω ≈ 0.378 rad/s

Now we can calculate the final kinetic energy:

KE_f = 0.5Iω^2
KE_f = 0.5(4.6 kg m^2)(0.378 rad/s)^2

KE_f ≈ 0.319 J

Therefore, the initial kinetic energy (KE_i) is 0 J and the final kinetic energy (KE_f) is approximately 0.319 J.