A circular disk of mass 0.2kg and radius 33cm, initially not rotating, slips down a thin spindle onto a turntable (disk) of mass 1.7kg and the same radius, rotating freely at 4.4rad/s.
a) Find the new angular velocity of the combination
b) The change in the kinetic energy?
c) If the motor is switched on after the disk has landed, what is the constant torque needed to regain the original speed in 3.3s?
To find the answers to these questions, we will need to use the principle of conservation of angular momentum and the formula for rotational kinetic energy.
a) To find the new angular velocity of the combination, we can use the principle of conservation of angular momentum, which states that the initial angular momentum of a system will be equal to the final angular momentum of the system.
The initial angular momentum of the circular disk can be calculated as the product of its moment of inertia and its initial angular velocity. The moment of inertia of a circular disk is given by the formula I = (1/2) * m * r^2, where m is the mass of the disk and r is its radius. In this case, the mass of the disk is 0.2kg and the radius is 33cm (or 0.33m).
So the initial angular momentum of the disk is I_disk * ω_initial = (1/2) * 0.2kg * (0.33m)^2 * 0, where ω_initial is the initial angular velocity of the disk (which is zero).
The initial angular momentum of the turntable is given by I_turntable * ω_initial, where I_turntable = (1/2) * 1.7kg * (0.33m)^2 (since it has the same mass and radius as the disk), and ω_initial is the initial angular velocity of the turntable, which is 4.4 rad/s.
Since there is no initial angular momentum for the disk (ω_initial = 0), the total initial angular momentum is simply the initial angular momentum of the turntable.
The final angular momentum of the system is the sum of the final angular momentum of the disk and the turntable. Let's say ω_final is the final angular velocity of the combination of the disk and the turntable.
So, the final angular momentum of the disk is I_disk * ω_final, and the final angular momentum of the turntable is I_turntable * ω_final.
Since the initial and final angular momenta are equal, we can set them equal and solve for ω_final:
I_turntable * ω_initial = I_disk * ω_final + I_turntable * ω_final
(1/2) * 1.7kg * (0.33m)^2 * 4.4 rad/s = (1/2) * 0.2kg * (0.33m)^2 * ω_final + (1/2) * 1.7kg * (0.33m)^2 * ω_final
Simplifying this equation will give you the value of ω_final, which is the new angular velocity of the combination.
b) To find the change in kinetic energy, we can use the formula for rotational kinetic energy, which is given by K = (1/2) * I * ω^2, where K is the kinetic energy, I is the moment of inertia, and ω is the angular velocity.
The initial kinetic energy of the system is the sum of the kinetic energies of the disk and the turntable. Since the disk is initially not rotating, its initial kinetic energy is zero. The initial kinetic energy of the turntable can be calculated using the formula K_turntable = (1/2) * I_turntable * ω_initial^2.
The final kinetic energy of the system is the sum of the kinetic energies of the disk and the turntable after they have come to rotational equilibrium. The kinetic energy of the disk can be calculated using the formula K_disk = (1/2) * I_disk * ω_final^2. The kinetic energy of the turntable can be calculated using the formula K_turntable = (1/2) * I_turntable * ω_final^2.
The change in kinetic energy is the final kinetic energy minus the initial kinetic energy.
c) To find the constant torque needed to regain the original speed in 3.3s, we can use the formula for torque, which is given by τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The angular acceleration α can be calculated using the formula α = (ω_final - ω_initial) / t, where ω_initial is the initial angular velocity, ω_final is the final angular velocity, and t is the time interval.
Once we have the angular acceleration α, we can calculate the torque τ using the formula τ = I * α, where I is the moment of inertia.
Let's calculate each of these values step by step.