Question 20.17: A turntable which can be modelled as a disc of radius 0.2 m ...
A turntable which can be modelled as a disc of radius 0.2 m and mass 5 kg is spinning horizontally about an axis through its centre 3 times a second. An object of mass 2 kg is gently placed on it 0.15 m from the centre and remains in place through friction. How fast does the turntable now rotate?
The angular momentum of the system as a whole is conserved.
The moment of inertia of the turntable is \frac{1}{2} × 5 × 0.2² = 0.1 kg m².
The turntable is rotating at 3 × 2π = 6πrad s^{-1}.
So the initial angular momentum is 0.1 × 6π = 0.6π (the object is initially at rest).
If the final angular velocity is ω, the angular momentum of the turntable is 0.1ω.
The speed of the object is 0.15ω so its angular momentum about the axis is
mass × tangential speed × distance from axis = 2 × 0.15ω × 0.15
= 0.045ω.
Hence the total final angular momentum is 0.145ω.
0.145ω = 0.6π
⇒ ω = 4.14π rad s^{-1}
which is 2.1 rotations per second.