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 need to use the conservation of angular momentum and the conservation of kinetic energy. Let's break down each part of the problem step by step.
A. Calculate the total moment of inertia before and after.
Before the student pulls the objects closer to the rotation axis, the moment of inertia can be calculated using the formula:
I_before = I_student + I_disk.
Given:
I_student = 0 (since the arms are extended horizontally)
I_disk = 3.00 kgm^2
Substituting the values into the equation:
I_before = 0 + 3.00 kgm^2 = 3.00 kgm^2.
After the student pulls the objects closer to the rotation axis, the moment of inertia can be calculated using a similar formula:
I_after = I_student + I_disk.
Given:
I_student = 4.00 kgm^2 (2 objects, each with a 2 kg mass)
I_disk = 3.00 kgm^2 (assuming the disk moment of inertia is constant)
Substituting the values into the equation:
I_after = 4.00 kgm^2 + 3.00 kgm^2 = 7.00 kgm^2.
Therefore, the total moment of inertia before is 3.00 kgm^2, and after is 7.00 kgm^2.
B. Calculate the new angular velocity after.
To find the new angular velocity after the objects are pulled closer, we can use the conservation of angular momentum equation:
I_before * ω_before = I_after * ω_after.
Given:
I_before = 3.00 kgm^2
I_after = 7.00 kgm^2
ω_before = 0.7 rad/sec (initial angular velocity)
Substituting the values into the equation:
3.00 kgm^2 * 0.7 rad/sec = 7.00 kgm^2 * ω_after.
Simplifying the equation:
0.7 rad/sec = 7.00 kgm^2 * ω_after / 3.00 kgm^2.
Solving for ω_after:
ω_after = (0.7 rad/sec) * (3.00 kgm^2) / (7.00 kgm^2).
ω_after ≈ 0.3 rad/sec.
Therefore, the new angular velocity after pulling the objects closer is approximately 0.3 rad/sec.
C. The K.E before and after.
The kinetic energy (K.E) can be calculated using the formula:
K.E = (1/2) * I * ω^2.
Before the objects are pulled closer:
K.E_before = (1/2) * 3.00 kgm^2 * (0.7 rad/sec)^2.
After the objects are pulled closer:
K.E_after = (1/2) * 7.00 kgm^2 * (0.3 rad/sec)^2.
Calculating the kinetic energy values:
K.E_before ≈ 0.735 J
K.E_after ≈ 0.315 J
Therefore, the kinetic energy before is approximately 0.735 J, and after is approximately 0.315 J.
D. How much energy was lost in the process.
To determine the energy lost in the process, we can calculate the difference between the initial and final kinetic energies:
Energy_lost = K.E_before - K.E_after.
Substituting the calculated values:
Energy_lost = 0.735 J - 0.315 J.
Energy_lost ≈ 0.42 J.
Therefore, approximately 0.42 Joules of energy were lost in the process.
To solve this problem, we can use the principle of conservation of angular momentum and the formula for rotational kinetic energy.
A. The total moment of inertia before and after can be calculated by summing the moment of inertia of the student plus the disk with the moment of inertia of the objects:
Total moment of inertia before (I_before) = moment of inertia of student + moment of inertia of disk
Total moment of inertia after (I_after) = moment of inertia of student + moment of inertia of disk + 2 * moment of inertia of objects
Given that the moment of inertia of the student plus the disk is 3.00 kgm^2, we have:
I_before = 3.00 kgm^2
I_after = 3.00 kgm^2 + 2 * (4 kg * (0.32 m)^2)
B. The new angular velocity after (ω_after) can be calculated using the principle of conservation of angular momentum:
I_before * ω_before = I_after * ω_after
Substituting the values, we have:
3.00 kgm^2 * (0.7 rad/sec) = I_after * ω_after
Solving for ω_after will give us the new angular velocity.
C. The kinetic energy before (KE_before) can be calculated using the formula for rotational kinetic energy:
KE_before = 0.5 * I_before * (ω_before)^2
The kinetic energy after (KE_after) can be calculated using the same formula but with the new angular velocity:
KE_after = 0.5 * I_after * (ω_after)^2
D. The energy lost in the process can be calculated by finding the difference between the initial kinetic energy and the final kinetic energy (assuming energy is conserved):
Energy lost = KE_before - KE_after
Let's calculate the values step-by-step:
A. Calculate the total moment of inertia:
Total moment of inertia before:
I_before = 3.00 kgm^2
Total moment of inertia after:
I_after = 3.00 kgm^2 + 2 * (4 kg * (0.32 m)^2)
B. Calculate the new angular velocity after:
Using the principle of conservation of angular momentum:
I_before * ω_before = I_after * ω_after
Solve for ω_after.
C. Calculate the kinetic energy before and after:
Kinetic energy before:
KE_before = 0.5 * I_before * (ω_before)^2
Kinetic energy after:
KE_after = 0.5 * I_after * (ω_after)^2
D. Calculate the energy lost:
Energy lost = KE_before - KE_after
Let's substitute the values and calculate the answers step-by-step.