NCERT Class 11 Physics Solutions: Chapter 7 – System of Particles and Rotational Motion Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 7.11:

Torques of equal magnitude is applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

Answer:

Let and r be the respective masses of the hollow cylinder and the solid sphere.

The moment of inertia of the hollow cylinder about its standard axis,

The moment of inertia of the solid sphere about an axis passing through its centre,

We have the relation:

Where,

Angular acceleration

Torque

Moment of inertia

For the hollow cylinder,

For the solid sphere,

As an equal torque is applied to both the bodies,

Now, using the relation:

Where,

Initial angular velocity

Time of rotation

Final angular velocity

For equal and , we have:

From equations and we can write:

Hence, the angular velocity of the solid sphere will be greater than that of the hollow cylinder.

Question 7.12:

A solid cylinder of mass rotates about its axis with angular speed . The radius of the cylinder is . What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

Answer:

Mass of the cylinder,

Angular speed,

Radius of the cylinder,

The moment of inertia of the solid cylinder:

Kinetic energy

Angular momentum,

Question 7.13:

(a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of . How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to times the initial value? Assume that the turntable rotates without friction.

(b) Show that the child՚s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do your account for this increase in kinetic energy?

Answer:

(a)

Explanation:

Initial angular velocity,

Final angular velocity

The moment of inertia of the boy with stretched hands

The moment of inertia of the boy with folded hands

The two moments of inertia are related as:

Since no external force acts on the boy, the angular momentum L is a constant.

Hence, for the two situations, we can write:

(b)

Explanation:

Final K. E. Initial K. E.

Final kinetic rotation,

Initial kinetic rotation,

The increase in the rotational kinetic energy is attributed to the internal energy of the boy.