A student stands on a rotating disk holding 4-kg objects with each hand. hen his arms are extended horizontally, the object are at 0.8m from the axis of rotation and he rotates at 0.7 rad/sec. The moment of inertia of the student plus the disk is 3.00 kgm^2 and is assumed to be constant. The student the pulls the objects horizontally to 0.32 m from the rotation axis.
A. Calculate the total moment of inertia before and after.
B. Calculate the new angular velocity after.
C. The K.E before and after.
D. How much energy was lost in the process.
To solve this problem, we will need to use the conservation of angular momentum. The formula for angular momentum is given by:
Angular Momentum = Moment of Inertia * Angular Velocity
Let's solve each part of the problem step by step:
A. Calculate the total moment of inertia before and after:
The total moment of inertia before is given by the moment of inertia of the student plus the disk. This is already provided as 3.00 kgm^2.
The total moment of inertia after can be calculated using the equation for moment of inertia of two point masses rotating around an axis:
Moment of Inertia = (mass1 * distance1^2) + (mass2 * distance2^2)
Before the objects were pulled in, the total moment of inertia was given as 3.00 kgm^2. After pulling the objects horizontally to 0.32 m from the rotation axis, we can calculate the new moment of inertia as:
(4 kg * 0.8 m^2) + (4 kg * 0.8 m^2) = 6.4 kgm^2
Therefore, the total moment of inertia after the objects were pulled in is 6.4 kgm^2.
B. Calculate the new angular velocity after:
The angular momentum before and after should remain the same since no net external torque acts on the system. So we can use the formula:
Angular Momentum before = Angular Momentum after
Moment of Inertia before * Angular Velocity before = Moment of Inertia after * Angular Velocity after
We know that the Moment of Inertia before is 3.00 kgm^2, Moment of Inertia after is 6.4 kgm^2, and Angular Velocity before is 0.7 rad/sec. We can rearrange the formula to solve for Angular Velocity after:
Angular Velocity after = (Moment of Inertia before * Angular Velocity before) / Moment of Inertia after
Angular Velocity after = (3.00 kgm^2 * 0.7 rad/sec) / 6.4 kgm^2
Solving this equation, we get the new angular velocity after pulling the objects in.
C. The K.E before and after:
The kinetic energy before and after can be calculated using the formula:
Kinetic Energy = 0.5 * Moment of Inertia * (Angular Velocity)^2
For the K.E before, we plug in the values of Moment of Inertia before (3.00 kgm^2) and Angular Velocity before (0.7 rad/sec) into the formula.
For the K.E after, we plug in the values of Moment of Inertia after (6.4 kgm^2) and Angular Velocity after (obtained from part B) into the formula.
D. How much energy was lost in the process:
The energy lost in the process can be calculated by subtracting the K.E after from the K.E before.
This will yield the amount of energy lost in the process of pulling the objects closer to the rotation axis.