Consider a system of two particles having masses ${\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2$. If the particle of mass ${\mathrm{m}}_1$ is pushed towards the center of mass of particles through a distance, by what distance would be the particle of the mass ${\mathrm{m}}_2$ move so as to keep the center of mass of particles at the original position?

1.  $\frac{{\mathrm{m}}_1}{{\mathrm{m}}_1 + {\mathrm{m}}_2}\mathrm{d}$

2.  $\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\mathrm{d}$

3.  d

4.  $\frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}\mathrm{d}$

A flywheel is in the form of a uniform circular disc of radius 1 m and mass 2 kg. The work which must be done on it to increase its frequency of rotation from 5 rev ${\mathrm{s}}^{-1}$ to 10 rev ${\mathrm{s}}^{-1}$ is approximately

1.  1.5 x ${10}^2$ J

2.  3.0 x ${10}^2$ J

3.  1.5 x ${10}^3$ J

4.  3.0 x ${10}^3$ J

A constant torque acting on a uniform circular wheel changes its angular momentum from ${\mathrm{A}}_0$ to $4 {\mathrm{A}}_0$ in 4s. The magnitude of this torque is 

1. $\frac{3{\mathrm{A}}_0}{4}$

2. $4 {\mathrm{A}}_0$

3. ${\mathrm{A}}_0$

4. $12 {\mathrm{A}}_0$

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s${}^{-1}$. Disc D${}_2$ has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s${}^{-1}$) of the system is

1. 60

2. 100

3. 120

4. 40

A body rolls down an inclined plane without slipping. The fraction of total energy associated with its rotation will be

$1. {\mathrm{K}}^2+ {\mathrm{R}}^2$
$2. \frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$
$3. \frac{{\mathrm{K}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$
$4. \frac{{\mathrm{R}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$

Where k is radius of gyration of the body about an axis passing through centre of mass and R is the radius of the body. 

In the following figure, a weight W is attached to a string wrapped round a solid cylinder of mass M mounted on a frictionless horizontal axle at O.

If the weight starts from rest and falls a distance h, then its speed at that instant is

1. Proportional to R

2. Proportional to $\frac{1}{R}$

3. Proportional to $\frac{1}{R^2}$

4. Independent of R

A solid cylinder rolls down an inclined plane that has friction sufficient to prevent sliding. The ratio of rotational energy to total kinetic energy is

1.  $\frac{1}{2}$

2.  $\frac{1}{3}$

3.  $\frac{2}{3}$

4.  $\frac{3}{4}$

A swimmer while jumping into water from a height easily forms a loop in the air, if

1. He pulls his arms and legs in

2. He spreads his arms and legs

3. He keeps himself straight

4. None of the above

Assertion : A ladder is more apt to slip when you are high up on it than when you just begin to climb

Reason : At the high point on a ladder, the torque is large and on climbing up the torque is small

An inclined plane makes an angle of $30°$ with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to

1.  $\frac{g}{3}$

2.  $\frac{5g}{7}$

3.  $\frac{2g}{3}$

4.  $\frac{5g}{14}$