Conservation of angular momentum and consequential changes in rotational kinetic energy

Question

Suppose for example consider the following abstract example, a circular disk that has several objects joined to it via short weak thin strings is made to rotate about an axis with an initial angular velocity = w₀

A pole attached to the axis and extended further has a very sharp knife oriented in the plane of rotation that starts sliding down simultaneously with the beginning of rotation

When it reaches the required height, the strings begin to get cut and the particles fly off with respective linear momentum and kinetic energies corresponding to the linear velocities they had at the time of detachment (v= w₀r) Thus the magnitude of kinetic energy for each individual particle remains the same (considering their initial moi = mr^2)

However at the instant that all the n objects have left the system the rotational kinetic energy of the disk that remains is more than the combined total kinetic energy earlier

E_i= (1/2)(I+ni)(w₀)^2

w_f= (I+ni)(w₀)/I ... (conserv. ang. momen.)

E_f= [((I+ni)(w₀))^2]/2I > E_i

How does this come about as there is no external force present to produce the necessary torque? Does the impact with the blade produce the required torque? Or have i misunderstood some concept?

Answer 1

The angular momentum of the system does not change, therefore no torque is required. The particles which fly off have angular momentum about the axis, and the disk has angular momentum about the axis. The total angular momentum is the same before and after the particles fly off. The angular speed of the disk does not change.

Your calculation is incorrect because you assumed that only the disk has angular momentum after the particles fly off. This is not correct. The particles, which have linear momentum, also have angular momentum about the axis of the disk. Linear and angular momentum are not mutually exclusive : an object can have both.

Energy is also conserved : the particles and the disk each have the same linear or rotational speed (and therefore also the same kinetic energy) before and after the particles fly off.

A similar question is : What happens to the speed (or momentum) of a truck as a load of objects fall off it on the highway? The objects have the same speed as the truck before and after falling off, so there is no change in speed or momentum of the truck. See Will the momentum be conserved in this scenario?