A turntable with a moment of inertia of 0.029 kg*m^2 rotates feely at 2.8 rad/s. A circular disk of mass 270 g and diameter 20 cm, and initially not rotating, slips down a spindle and lands on the turntable.
a) Find the new angular speed.
b) What is the change in kinetic energy?
a) Angular momentum about the spindle is conserved.
Iturntable*w1 = (Iturntable+Icd)w2
Solve for the final angular velocity w2.
b) (1/2)*Iturntable*w1^2 - (1/2)(Iturntable + Icd)*w2^2
The formula for the compact disc moment of inertia Icd is:
Icd = (1/2)Mcd*Rcd^2
To find the new angular speed (ωf) of the turntable, we need to apply the law of conservation of angular momentum. The initial angular momentum (Li) of the system, consisting of the turntable and the disk, is equal to the final angular momentum (Lf) of the system.
Angular momentum (L) is given by the product of moment of inertia (I) and angular speed (ω):
L = I * ω
a) Since the only change in the system is the addition of the disk to the turntable, we can express the initial angular momentum as:
Li = I_turntable * ω_turntable
Since the disk initially has no angular speed, its initial angular momentum is zero:
Li_disk = I_disk * ω_disk = 0
Hence, the initial angular momentum of the system is the initial angular momentum of the turntable:
Li = I_turntable * ω_turntable
After the disk lands on the turntable, the combined system will have a new angular momentum equal to:
Lf = (I_turntable + I_disk) * ωf
According to the law of conservation of angular momentum, Li = Lf, so:
I_turntable * ω_turntable = (I_turntable + I_disk) * ωf
Rearranging the equation to solve for ωf:
ωf = (I_turntable * ω_turntable) / (I_turntable + I_disk)
Hence, to find the new angular speed (ωf) of the turntable, we need to substitute the given values for I_turntable, ω_turntable, I_disk, and perform the calculation.
b) To find the change in kinetic energy, we need to calculate the initial kinetic energy (Ki) and the final kinetic energy (Kf) of the system. The change in kinetic energy (ΔK) is the difference between the final and initial kinetic energies:
ΔK = Kf - Ki
The initial kinetic energy of the system is:
Ki = 1/2 * I_turntable * ω_turntable^2
The final kinetic energy of the system is:
Kf = 1/2 * (I_turntable + I_disk) * ωf^2
Substituting the known values for I_turntable, ω_turntable, I_disk, and ωf into the equations and performing the calculations will give us the final answer.
Note: The moment of inertia (I) of a circular disk is given by I = (1/4) * m * r^2, where m is the mass of the disk and r is its radius.