Particles of masses m, 2m, 3m, ... , nm are placed on the same line at distances L, 2L, 3L, ... , nL from O. The distance of centre of mass from O is

A ball of radius R rolls without slipping. Find the fraction of total energy associated with its rotational energy, if the radius of the gyration of the ball about an axis passing through its centre of mass is K.

• $\frac{{\mathrm{K}}^2}{{\mathrm{K}}^2 + {\mathrm{R}}^2}$

• $\frac{{\mathrm{R}}^2}{{\mathrm{K}}^2 + {\mathrm{R}}^2}$

• $\frac{{\mathrm{K}}^2 + {\mathrm{R}}^2}{{\mathrm{R}}^2}$

• $\frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$

A solid sphere of mass M and radius 2 R rolls down an inclined plane of height h without slipping. The speed of its centre of mass when it reaches the bottom is

• $\sqrt{\frac{6}{7} \mathrm{gh}}$

• $\sqrt{3 \mathrm{gh}}$

• $\sqrt{\frac{10}{7} \mathrm{gh}}$

• $\sqrt{\frac{4}{3} \mathrm{gh}}$

A wheel starts rotating from rest at time t = 0 with a angular acceleration of 50 radians/s². The angular acceleration (α) decreases to zero value after 5 seconds. During this interval, a varies according to the equation

The angular velocity at t = 5 s will be

• 10 rad/s

• 250 rad/s

• 125 rad/s

• 100 rad/s

Assertion:  A solid sphere is rolling on a rough horizontal surface. Acceleration of contact point is zero. 

Reason:  A solid sphere can roll on the smooth surface.

• If both assertion and reason are true and reason is the correct explanation of assertion.

• If both assertion and reason are true but reason is not the correct explanation of assertion.

• If assertion is true but reason is false.

• If both assertion and reason are false.

76.

A sphere of mass 10 kg and radius 0.5 m rotates about a tangent. The moment of inertia of the sphere is

• 5 kg m²

• 2.7 kg m²

• 3.5 kg m²

• 4.5 kg m²

Assertion:  Value of radius of gyration of a body depends on axis of rotation. 

Reason:  Radius of gyration is root mean square distance of particle of the body from the axis of rotation.

• If both assertion and reason are true and reason is the correct explanation of assertion.

• If both assertion and reason are true but reason is not the correct explanation of assertion.

• If assertion is true but reason is false.

• If both assertion and reason are false.

Assertion:  A judo fighter in order to throw his opponent on to the mat tries to initially bend his opponent and then rotate him around his hip. 

Reason:  As the mass of opponent is brought closer to the fighter's hip, the force required to throw the opponent is reduced.

• If both assertion and reason are true and reason is the correct explanation of assertion.

• If both assertion and reason are true but reason is not the correct explanation of assertion.

• If assertion is true but reason is false.

• If both assertion and reason are false.

Two homogeneous spheres A and B of masses m and 2m having radii 2a and a respectively are placed in touch. The distance of centre of mass from first sphere is

• a

• 2a

• 3a

• none of these

Assertion:  A body subject to three concurrent forces may be in equilibrium. 

Reason: For equilibrium, the sum of all the concurrent forces acting at a point should be zero.

• If both assertion and reason are true and reason is the correct explanation of assertion.

• If both assertion and reason are true but reason is not the correct explanation of assertion.

• If assertion is true but reason is false.

• If both assertion and reason are false.