System of Particles and Rotational Motion

A uniform rod of length 200 cm and mass 500 g is balanced on a wedge placed at 40 cm mark. A mass of 2 kg is suspended from the rod at 20 cm and another unknown mass 'm' is suspended from the rod at 160 cm mark as shown in the figure. Find the value of 'm' such that the rod is in equilibrium. (g = 10 m/s2) [figure]

1. 112 kg
2. 12 kg
3. 13 kg
4. 16 kg

Two particles of mass 5 kg and 10 kg respectively are attached to the two ends of a rigid rod of length 1 m with negligible mass. The centre of mass of the system from the 5 kg particle is nearly at a distance of :

1. 50 cm
2. 67 cm
3. 80 cm
4. 33 cm

Find the torque about the origin when a force of 3 j N acts on a particle whose position vector is 2 k m.

1. 6 j N m
2. -6 i N m
3. 6 k N m
4. 6 i N m

A solid cylinder of mass 2 kg and radius 4 cm is rotating about its axis at the rate of 3 rpm. The torque required to stop after 2 revolutions is :

1. 2 10-3 N m
2. 2 10-6 N m
3. 2 106 N m
4. 12 10-4 N m

A disc of radius 2 m and mass 100 kg rolls on a horizontal floor. Its centre of mass has speed of 20 cm/s. How much work is needed to stop it?

1. 1 J
2. 2 J
3. 3 J
4. 30 kJ

A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy (Kt) as well as rotational kinetic energy (Kr) simultaneously. The ratio Kt : (Kt + Kr) for the sphere is

1. 7 : 10
2. 5 : 7
3. 10 : 7
4. 2 : 5

A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?

1. Angular velocity
2. Moment of inertia
3. Rotational kinetic energy
4. Angular momentum

Three objects, A : (a solid sphere), B : (a thin circular disk) and C : (a circular ring), each have the same mass M and radius R. They all spin with the same angular speed about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation

1. WC > WB > WA
2. WA > WB > WC
3. WB > WA > WC
4. WA > WC > WB

The moment of the force, F = 4 i + 5 j - 6 k at (2, 0, −3), about the point (2, −2, −2), is given by

1. -8 i - 4 j - 7 k
2. -4 i - j - 8 k
3. -7 i - 8 j - 4 k
4. -7 i - 4 j - 8 k

A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N?

1. 0.25 rad s-2
2. 25 rad s-2
3. 5 m s-2
4. 25 m s-2