A solid cylinder and a solid sphere, both having the same mass and radius, are released from a rough inclined plane. Both roll without slipping. Then,

1.  the force of friction that acts on the two is the same

2.  the force of friction is greater in the case of a sphere than in a cylinder

3.  the force of friction is greater in the case of a cylinder than in a sphere

4.  the force of friction will depend on the nature of the surface of the body that is moving and that of the inclined surface and is independent of the shape and size of the moving body

A homogeneous disc of mass 2 kg and radius 15 cm is rotating about its axis (which is fixed) with an angular velocity of 4 rad/sec. The linear momentum Of the disc is 

1.  1.2 kg m/sec 

2.  1.0 kg m/sec 

3.  0.6 kg m/sec 

4.  None of these

A rigid body rotates about a fixed axis with a variable angular velocity equal to \(\alpha -\beta t\), at the time \(t\), where \(\alpha , \beta\) are constants. The angle through which it rotates before it stops is:

For the uniform T shaped structure, with mass \(3~ \text M\), the moment of inertia about an axis normal to the plane and passing through O would be

            

1. \({ 2 \over 3} \text {MI}^2\)
2. \(\text {MI}^2\)
3. \({ MI^2 \over 3}\)
4. None of these 

A rigid body rotates with an angular momentum L. If its rotational kinetic energy is made 4 times, its angular momentum will become

1.  4L

2.  16L

3.  $\sqrt{2} \mathrm{L}$

4.  2L

Four particles of mass \(m_1 = 2m\), \(m_2=4m\), \(m_3 =m \), and \(m_4\) are placed at the four corners of a square. What should be the value of \(m_4\) so that the centre of mass of all the four particles is exactly at the centre of the square?

Angular momentum of a body is defined as the product of

1.  Mass and angular velocity

2.  Centripetal force and radius

3.  Linear velocity and angular velocity

4.  Moment of inertia and angular velocity

Two masses ${\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2, {\mathrm{m}}_1 > {\mathrm{m}}_2$ are connected to the ends of massless rope and allowed to move as shown in the figure. The acceleration of the centre of mass assuming pulley is massless and frictionless, is

1.  \(\frac{m_{1}   -   m_{2}}{m_{1}   +   m_{2}} g\)

2.  0

3.  \({m_1 -m_2^2 \over m_1 + m_2 } g \)

4.  \({m_1 +m_2^2 \over m_1 - m_2 } g \)

The speed of a uniform spherical shell after rolling down an inclined plane of vertical height h from rest is:

1.  \(\sqrt{\frac{10   g h}{7}}\)

2.    \(\sqrt{\frac{6   g h}{5}}\)

3.  \(\sqrt{\frac{4   g h}{5}}\)

4.  \(\sqrt{2   g h}\)

A uniform rod of mass m is bent into the form of a semicircle of radius R. The moment of inertia of the rod about an axis passing thorugh A and Perpendicular to the plane of paper is

1. $\frac{2}{3}M{R}^2$

2. $M{R}^2$

3. 2$M{R}^2$

4. $\frac{3}{2}M{R}^2$