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 calculate the initial and final kinetic energy of the system, we need to apply the principle of conservation of angular momentum.
1. Find the initial angular momentum:
The angular momentum of the system is given by the product of the moment of inertia (I) and angular velocity (ω). Since the student is initially at rest, the initial angular momentum is zero.
2. Find the final angular momentum:
The final angular momentum is given by the product of the moment of inertia (I) and the final angular velocity (ωf). To find ωf, we can use the conservation of angular momentum equation:
Ii * ωi = If * ωf, where Ii and ωi are the initial values, and If and ωf are the final values.
3. Calculate the final angular velocity:
Rearranging the equation from step 2, we have:
ωf = (Ii * ωi) / If
4. Find the initial and final kinetic energy:
The initial kinetic energy is zero since the student is at rest. The final kinetic energy is given by the equation:
Kf = (1/2) * If * ωf^2, where Kf is the final kinetic energy, If is the moment of inertia of the system, and ωf is the final angular velocity calculated in step 3.
Now, let's plug in the given values and calculate the final kinetic energy.
Given:
Moment of inertia (I) = 4.6 kg m^2
Mass of thrown object (m) = 1.5 kg
Speed of thrown object (v) = 2.9 m/s
Distance from axis of rotation (r) = 0.40 m
Using the formula for moment of inertia, I = m * r^2, we can convert the mass and distance into the moment of inertia of the thrown object.
Moment of inertia of thrown object (It) = m * r^2
= 1.5 kg * (0.40 m)^2
= 0.24 kg m^2
Now, we can substitute the values into the equations above to calculate the final angular velocity (ωf) and the final kinetic energy (Kf).
ωf = (Ii * ωi) / If
= (0 * 0) / (4.6 kg m^2 + 0.24 kg m^2)
= 0 rad/s
Kf = (1/2) * If * ωf^2
= (1/2) * (4.6 kg m^2 + 0.24 kg m^2) * (0 rad/s)^2
= 0 J
Therefore, the initial kinetic energy is zero (since the student is at rest), and the final kinetic energy is also zero (since the final angular velocity is zero).