Question

A student sits on a freely rotating stool holding two weights, each of mass 4 kg.. When his arms are extended horizontally, the weights are
1.1 m from the axis of rotation and he rotates with an angular speed of 0.9 rad/s. The moment of inertia of the student plus stool is 3.0 kg-m2 and is assumed to be constant. The student pulls the weights inward horizontally to a position 0.4 m from the rotation axis.
Find the new angular speed of the student.
Find the kinetic energy of the rotating system before he pulls the weights inward.
Find the kinetic energy of the rotating system after he pulls the weights inward.

Answer 1

Concepts and reason

The concepts used to solve this problem, are the conservation of angular momentum and the rotational kinetic energy.

First, use the angular momentum conservation to find the new angular speed of the student.

Then, use the relation between the rotational kinetic energy, angular speed, and the moment of inertia to find the energy of the rotating system.

Fundamentals

The angular momentum conservation states that “total angular momentum, both the magnitude and the direction of an isolated system remains constant”.

Expression for the total angular momentum of the system is,

Here, is the initial moment of inertia, is the initial angular speed, is the final moment of inertia, and is the final angular speed.

The expression for the moment of inertia of the weight is as follows:

$zum=7 $

Here, is the moment of inertia of the weight, is the mass, and is the distance of the object from the axis of rotation.

Expression for the rotational kinetic energy is as follows:

Here, is the rotational kinetic energy, is the moment of inertia, and is the angular speed.

(1)

The expression for the moment of inertia of the weight is as follows:

$zum=7 $

The expression for the total moment of inertia of the system when the arms are stretched is as follows:

Here, is the total moment of inertia of the system when the arms are stretched, is the distance of weight when the arms are stretched and is the moment of inertia of the student and the stool.

Substitute for , for, and for .

The expression for the total moment of inertia of the system when the arms are closed inward is,

Here, is the total moment of inertia of the system when the arms are closed inward and is the distance of weight when the arms are closed inward.

Substitute for, for, and for

Expression for the total angular momentum of the system is as follows:

Rearrange the above expression for

Substitute for, for, and for

Therefore, the new angular speed of the student when the arms are closed inward is.

(2)

Expression for the rotational kinetic energy is,

Expression for the kinetic energy of the rotating system before pulls the weight inward is,

Here, is the kinetic energy of the rotating system before pulls the weight inward.

Substitute for and for.

Therefore, the rotational kinetic energy before pulls the weight inward is.

(3)

Expression for the rotational kinetic energy is as follows:

Expression for the kinetic energy of the rotating system after pulls the weight inward is as follows:

Here, is the kinetic energy of the rotating system after pulls the weight inward.

Substitute for and for

Therefore, the rotational kinetic energy after pulls the weight inward is.

Ans: Part 1

Thus, the new angular speed of the student when the arms are closed is.