A uniform disk of mass 2kg and radius 10.0cm is spinning on a fixed frictionless bearing at 3 revolutions/second. A think cylindrical shell of mass 2kg and radius 10cm is carefully dropped onto the spinning disk so that they end up spinning together, with their edges coinciding.
a) How fast (in revolutions/second) are they spinning after the drop?
b) Is kinetic energy conserved is this process? If not, by what percentage has it changed?
To solve this problem, we need to apply the law of conservation of angular momentum.
Angular momentum (L) is given by the formula: L = Iω, where I is the moment of inertia and ω is the angular velocity.
The moment of inertia for a uniform disk is given by: I_disk = (1/2) * M * R^2, where M is the mass of the disk and R is its radius.
a) Before the drop:
The initial angular momentum of the spinning disk is given by L_initial = I_disk * ω_initial.
Substituting the values, we have L_initial = (1/2) * (2 kg) * (0.1 m)^2 * (3 rev/s).
After the drop:
The final angular momentum of the system (disk + shell) is given by L_final = (I_disk + I_shell) * ω_final.
The moment of inertia for a cylindrical shell is given by: I_shell = M_shell * R^2. In this case, M_shell = 2 kg (given), and R = 0.1 m.
Since the disk and shell are spinning together, their final angular velocities will be the same, denoted as ω_final.
To find ω_final, we equate the initial angular momentum (L_initial) and the final angular momentum (L_final):
L_initial = L_final
(1/2) * (2 kg) * (0.1 m)^2 * (3 rev/s) = ((1/2) * (2 kg) * (0.1 m)^2 + (2 kg) * (0.1 m)^2) * ω_final
Simplifying the equation, we can solve for ω_final:
(1/2) * (2 kg) * (0.1 m)^2 * (3 rev/s) = (1/2) * (2 kg) * (0.1 m)^2 * ω_final + (2 kg) * (0.1 m)^2 * ω_final
(1/2) * (2 kg) * (0.1 m)^2 * (3 rev/s) = (1/2) * (2 kg) * (0.1 m)^2 * ω_final + (2 kg) * (0.1 m)^2 * ω_final
(3 rev/s) = ω_final + ω_final
(3 rev/s) = 2ω_final
ω_final = (3 rev/s) / 2
ω_final = 1.5 rev/s
Hence, after the drop, the disk and the cylindrical shell will be spinning together at a rate of 1.5 revolutions/second.
b) To determine if kinetic energy is conserved, we need to compare the initial kinetic energy (KE_initial) with the final kinetic energy (KE_final).
The kinetic energy for rotating objects is given by: KE = (1/2) * I * ω^2.
Before the drop, the initial kinetic energy (KE_initial) of the spinning disk is given by:
KE_initial = (1/2) * I_disk * ω_initial^2
After the drop, the final kinetic energy (KE_final) of the system is given by:
KE_final = (1/2) * (I_disk + I_shell) * ω_final^2
To determine if kinetic energy is conserved, we compare KE_initial with KE_final:
If KE_initial = KE_final, then kinetic energy is conserved.
Using the given values and the calculated values for angular velocities from the previous calculations, substitute them into the equations to determine KE_initial and KE_final.
Evaluate the kinetic energy conservation by comparing the calculated values.
If the kinetic energy is not conserved, to determine the percentage change, use the formula:
Percentage Change = [ (KE_final - KE_initial) / KE_initial ] * 100
Calculate the percentage change to determine how much the kinetic energy has changed.