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If a gymnast, sitting on a rotating stool, with his arms outstretched, suddenly lowers his hands

the angular velocity increases

his moment of inertia increases

the angular velocity stays constant

the angular momentum increases

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Explanation

When mass is placed nearer the axis of rotation, M.I. decreases, angular velocity increases.

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Moment of inertia of a uniform circular disc about a diameter is I. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be

5 I

3 I

6 I

4 I

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Explanation

M.I. of a disc about any diameter = $\frac{{MR}^{2}}{4} = I$

M.I. of the disc about an axis passing through a point on its rim and perpendicular to its plane

$= \frac{3}{2} {MR}^{2} = \frac{3}{2} \times 4 I = 6 I$

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A disk and a sphere of the same radius but different masses roll off on two inclined planes Of the same altitude and length Which one of the two objects gets to the bottom of the plane first

Disk

Sphere

Both reach at the same time

Depends on their masses

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Explanation

(b)

$\begin{aligned} a = \frac{gsin θ}{1 + \frac{K^{2}}{R^{2}}} \\ for disc; \frac{K^{2}}{R^{2}} = \frac{1}{2} = 0 .5 \\ for sphere; \frac{K^{2}}{R^{2}} = \frac{2}{5} = 0 .4 \end{aligned}$

a(sphere)>a(disc)

$∴$ sphere reaches first

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A ball hits the floor and rebounds after an inelastic collision. In this case

the momentum of the ball just after the collision is the same as that just before the collision

the mechanical energy of the ball remains the same in the collision

the total momentum of the ball and the earth is conserved

the total energy of the ball and the earth is conserved

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Explanation

Net force on ball and earth system is zero.

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The rotational KE of a body is E and its moment of inertia is I. The angular momentum is

$2 \sqrt{(EI)}$

$\sqrt{(2 EI)}$

$\frac{E}{I}$

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Explanation

$\begin{aligned} E & = \frac{1}{2} {Iω}^{2} \\ ω & = \sqrt{\frac{2 E}{I}} \\ Now, L & = Iω = \sqrt{2 EI} \end{aligned}$

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If rotational kinetic energy is 50 % of translational kinetic energy, then the body is

Ring

Cylinder

Hollow sphere

Solid sphere

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Explanation

For a cylinder, the rotational kinetic energy is half of its translational kinetic energy. This is because the moment of inertia of a cylinder about its axis is (1/2) MR^2, where M is the mass and R is the radius.

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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

zero

1/2 a

2 a

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Explanation

The center of mass of a system of particles moves as if the entire mass of the system were concentrated at that point and all external forces were applied there. For two identical particles, one at rest and the other with acceleration 'a', the center of mass acceleration is (a/2).

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

5 ${ms}^{- 2}$

$\frac{7}{25} {ms}^{- 2}$

$\frac{25}{7} {ms}^{- 2}$

$\frac{15}{7} {ms}^{- 2}$

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Explanation

$\begin{aligned} a & = \frac{gsin θ}{1 + \frac{k^{2}}{r^{2}}} = \frac{gsin 30^{\circ}}{1 + \frac{2}{5}} = \frac{10 \times \frac{1}{2}}{\frac{7}{5}} \\ = \frac{5}{7} \times 5 = \frac{25}{7} {ms}^{- 2} \end{aligned}$

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

8 rad/s

12 rad/s

24 rad /s

36 rad/s

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Explanation

$ω = \frac{d θ}{d t} = 6 t^{2}, A t t = 2 s ω = 6 {(2)}^{2} = 24 r a d / s$

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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

the angular velocity of the system decreases

the angular velocity of the system increases

the angular velocity of the system remains unchanged

the moment of inertia of the system decreases

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Explanation

When Ice melts, water tends to move away from the axis, M.I increases, by conservation of angular momentum, angular velocity decreases.

Choice A is correct.

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