A circular disk of mass 0.3 kg and radius 40 cm, initially not rotating, slips down a thin spindle onto a turntable (disk) of mass 1.7 kg and the same radius, rotating freely at 3.1 rad/s.

a) Find the new angular velocity of the combination;
A circular disk of mass 0.3 kg and radius 40 cm, initially not rotating, slips down a thin spindle onto a turntable (disk) of mass 1.7 kg and the same radius, rotating freely at 3.1 rad/s.

a) Find the new angular velocity of the combination;
rad/s

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 3.9 s?
N*m

To find the new angular velocity of the combination, we can use the principle of conservation of angular momentum. Angular momentum is given by the product of the moment of inertia and the angular velocity:

L = Iω

However, in this case, the initial angular momentum of the disk is zero since it is initially not rotating. So the angular momentum of the combination after the disk lands is just the angular momentum of the turntable:

L = I_turntable * ω_turntable

The moment of inertia of a disk is given by the formula:

I = (1/2) * m * r^2

where m is the mass of the object and r is the radius. Both the disk and the turntable have the same radius, so the moment of inertia of the turntable is:

I_turntable = (1/2) * m_turntable * r^2 = (1/2) * 1.7 kg * (0.4 m)^2

Now we can calculate the new angular velocity using the angular momentum conservation:

L = I_turntable * ω_turntable

ω_new = L / I_turntable

Substituting the values into the equation, we get:

ω_new = 0.0 kg.m^2/s / [(1/2) * 1.7 kg * (0.4 m)^2]

Calculate the value to find the new angular velocity of the combination in rad/s.