Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration a. The centre of mass has an acceleration

1. zero

2. \(\frac{a}{2}\)

3. a

4. 2a

A solid sphere rolls without slipping down a $30°$ inclined plane. If g = 10 $\mathrm{m}/{\mathrm{s}}^2$, the acceleration of the rolling sphere is

1.  5 ${\mathrm{ms}}^{-2}$

2.  $\frac{7}{25}{\mathrm{ms}}^{-2}$

3.  $\frac{25}{7}{\mathrm{ms}}^{-2}$

4.  $\frac{15}{7}{\mathrm{ms}}^{-2}$

If the equation for the displacement of a particle moving on a circular path is given by $\theta =2t^3+0.5$, where $\theta$ is in radian and t is in second, then the angular velocity of the particle after 2s is 

1. 8 rad/s

2. 12 rad/s

3. 24 rad /s

4. 36 rad/s

A pan containing a layer of uniform thickness of ice is placed on a circular turntable with its centre coinciding with the centre of the turn table. The turntable is now rotated at a constant angular velocity about a vertical axis passing through its centre and then driving is withdrawn. There is no friction between the table. As the ice melts

1.  the angular velocity of the system decreases

2.  the angular velocity of the system increases

3.  the angular velocity of the system remains unchanged

4.  the moment of inertia of the system decreases

The motion of planets in the solar system is an example of the conservation of

1.  mass

2.  Linear momentum

3.  Angular momentum

4.  Energy

A circular disc is to be made by using iron and aluminium, so that it acquires maximum moment of inertia about its geometrical axis.  It is possible with

1. iron and aluminium layers in alternate order

2. aluminium at interior and iron surrounding it

3. iron at interior and aluminium surrounding it

4. either '1' or '3'

Two Circular discs A and B are of equal masses and thicknesses but made of metal with densities ${\mathrm{d}}_{\mathrm{A}} \mathrm{and} {\mathrm{d}}_{\mathrm{B}}$ (${\mathrm{d}}_{\mathrm{A}} > {\mathrm{d}}_{\mathrm{B}}$). If their moments of inertia about an axis passing through their centers and perpendicular to circular faces be ${\mathrm{I}}_{\mathrm{A}} \mathrm{and} {\mathrm{I}}_{\mathrm{B}}$, then

1.  ${\mathrm{I}}_{\mathrm{A}} = {\mathrm{I}}_{\mathrm{B}}$

2.  ${\mathrm{I}}_{\mathrm{A}} > {\mathrm{I}}_{\mathrm{B}}$

3.  ${\mathrm{I}}_{\mathrm{A}}\hspace{0.17em}<\hspace{0.17em}{\mathrm{I}}_{\mathrm{B}}$

4.  ${\mathrm{I}}_{\mathrm{A}} \ge {\mathrm{I}}_{\mathrm{B}}$

A body is rolling without slipping on a horizontal surface and its rotational kinetic energy is equal to the translational kinetic energy. The body is 

1.  Disc

2.  Sphere

3.  Cylinder

4.  Ring

A wheel with a radius of \(20\) cm has forces applied to it as shown in the figure. The torque produced by the forces of \(4\) N at \(A\), \(8~\)N at \(B\), \(6\) N at \(C\), and \(9~\)N at \(D\), at the angles indicated, is:

1. \(5.4\) N-m anticlockwise

2. \(1.80\) N-m clockwise

3. \(2.0\) N-m clockwise

4. \(3.6\) N-m clockwise

The moment of inertia of a thin rectangular plate \(\text {ABCD}\) of uniform thickness about an axis passing through the centre \(\text O\) and perpendicular to the plane of the plate is 

1.  \(\text I_{1}   +   \text I_{2}\)

2.  \(\text I_{2}   +  \text I_{4}\)

3.  \(\text I_{1}   - \text   I_{3}\)

4.  \(\text I_{1}   +\text   I_{2  } +  \text I_{3}   + \text   I_{4}\)