# NCERT Class 11 Physics Solutions: Chapter 7 – System of Particles and Rotation Motion-Part 12

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Question 7.27:

Prove the result that the velocity of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height is given by .

Using dynamical consideration (i.e. by consideration of forces and torques). Note is the radius of gyration of the body about its symmetry axis, and is the radius of the body. The body starts from rest at the top of the plane.

A body rolling on an inclined plane of height ,is shown in the following figure:

Mass of the body

Radius of gyration of the body

Translational velocity of the body

Height of the inclined plane

Acceleration due to gravity

Total energy at the top of the plane,

Total energy at the bottom of the plane,

But,

From the law of conservation of energy, we have:

Hence, the given result is proved.

Question 7.28:

A disc rotating about its axis with angular speed is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is . What are the linear velocities of the points and on the disc shown in Figure? Will the disc roll in the direction indicated?

;

The disc will not roll

Explanation:

Angular speed of the disc

Using the relation for linear velocity,

For point A:

in the direction tangential to the right

For point B:

; in the direction tangential to the left

For point C:

in the direction same as that of

The directions of motion of points on the disc are shown in the following figure:

Since the disc is placed on a frictionless table, it will not roll. This is because the presence of friction is essential for the rolling of a body.