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QUESTION: 1

A spherical solid ball rolls on a table. What fraction of its total K.E. is rotational K.E. ?

Solution:

QUESTION: 2

A body is rolling down an inclined plane without slipping. How does the acceleration of the rolling body depend on its radius?

Solution:

The acceleration of a rolling body is independent of its mass and radius.

QUESTION: 3

Acceleration (a) of a body of radius R rolling down an inclined plane of inclination is given by

Solution:

QUESTION: 4

What is the moment of inertia of a solid sphere about its diameter?

Solution:

We have to consider solid sphere contribution of infinite small solid cylinder.

Icylinder = ½ MR²

dI = ½ dmr²

dm = pdV

dV = πr²dx

By substituting dV into dm

dm = pπr²dx

Now dm into dI

dI = ½ r²pπr²dx = ½ pπr⁴dx

Now, r² = R²-x²

dI = ½ pπ(R²-x²)²dx

Integrate both sides

I = ½ pπ 16/15 R²

P = M/V

P = M/4/3πR³ (density of sphere)

After substituting

I = 2/5MR^{2}

QUESTION: 5

What is the coefficient of friction when a cylinder rolls down without slipping on an inclined plane?

Solution:

QUESTION: 6

If a solid cylinder rolls up along a inclined plane with an initial velocity v. It will rise up to a height h equal to

Solution:

QUESTION: 7

What will be the speed of center of mass, if a solid sphere of radius 20 cm which rotates at 2 rps along a straight line?

Solution:

V = rw

V = 20 × 10^{-2} × 2π(2)

V = 2.51 m/s

QUESTION: 8

Two discs A and B are mounted on a vertical axle. Suppose the discs have moment of inertia as I and 2 I respectively about an axis. Disc A is given angular velocity ω, using the potential energy of a spring compressed by x_{1}. Similarly the potential energy of a spring compressed by x_{2} gives an angular velocity of 2ω to Disc B. Considering that both the discs rotate in same direction .Find the ratio x_{1}/ x_{2 }:

Solution:

QUESTION: 9

A small cylinder rolling with a velocity v along a horizontal surface encounters a smooth inclined surface. The height ‘h’ up to which the cylinder will ascend is

Solution:

A body can roll along a surface only if the surface is rough. The body will roll up to the foot of the inclined smooth surface. It will continue to spin with the angular speed it has acquired, and will slide up to a certain height, maintaining its spin motion throughout the smooth surface. Its translational kinetic energy alone is responsible for its upward motion along the smooth incline so that the height up to which it will rise is given by

QUESTION: 10

The kinetic energy of a rolling wheel is

Solution:

The total KE is the sum of translational KE and rotational KE.

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