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

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**Question 7.19**:

A hoop of radius weighs . It rolls along a horizontal floor so that its centre of mass has a speed of . How much work has to be done to stop it?

**Answer**:

Radius of the hoop,

Mass of the hoop,

Velocity of the hoop,

Total energy of the hoop Translational KE Rotational KE

Moment of inertia of the hoop about its centre,

But we have the relation,

The work required to be done for stopping the hoop is equal to the total energy of the hoop.

Required work to be done,

**Question 7.20**:

The oxygen molecule has a mass of and a moment of inertia of about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

**Answer**:

Mass of an oxygen molecule ,

Moment of inertia,

Velocity of the oxygen molecule,

The separation between the two atoms of the oxygen molecule

Mass of each oxygen atom

Hence, moment of inertia I, is calculated as:

It is given that:

**Question 7.21**:

A solid cylinder rolls up an inclined plane of angle of inclination At the bottom of the inclined plane the centre of mass of the cylinder has a speed of s.

(a) How far will the cylinder go up the plane?

(b) How long will it take to return to the bottom?

**Answer**:

A solid cylinder rolling up an inclination is shown in the following figure:

Initial velocity of the solid cylinder,

Angle of inclination,

Height reached by the cylinder

**(a)** How far will the cylinder go up the plane:

**Explanation**:

Energy of the cylinder at point

Energy of the cylinder at point

Using the law of conservation of energy, we can write:

Moment of inertia of the solid cylinder,

But we have relation,

In

Hence, the cylinder will travel m up the inclined plane.

**(b)** How long will it take to return to the bottom:

**Explanation**:

For radius of gyration , the velocity of the cylinder at the instance when it rolls back to the bottom is given by the relation:

For the solid cylinder ,

The time taken to return to the bottom is:

Therefore, the total time taken by the cylinder to return to the bottom is