A thin circular ring of mass M and radius r is rotating about its axis with constant angular velocity ω. Two objects each of mass m are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with angular velocity given by - 1. $\frac{2M\omega }{M+2m}$ 2. $\frac{\left(M+2m\right)\omega }{M}$ 3. $\frac{M\omega }{M+2m}$ 4. $\frac{\left(M+2m\right)\omega }{2m}$

Subtopic:  Angular Momentum |
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From a circular disc of radius R and mass 9M, a small disc of mass M and radius R / 3 is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is -

1. $M{R}^{2}$

2. 4$M{R}^{2}$

3. $\frac{4}{9}$$M{R}^{2}$

4. $\frac{40}{9}$$M{R}^{2}$

Subtopic:  Moment of Inertia |
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A solid cylinder and a hollow cylinder both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first-
1. Both together
2. Hollow cylinder
3. Solid cylinder
4. Both together only when angle of inclination of plane is 45º

Subtopic:  Rolling Motion |
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Given the following statements:
a. The centre of gravity (C.G.) of a body is the point at which the weight of the body acts.
b. If the earth is assumed to have an infinitely large radius, the centre of mass coincides with the centre of gravity.
c. To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G.
d. The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis.
Which one of the following pairs of statements is correct?
1. (a) and (b)
2. (b) and (c)
3. (c) and (d)
4. (d) and (a)

Subtopic:  Center of Mass |
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Three masses are placed on the x-axis : 300 g at origin, 500 g at x = 40 cm and 400 g at x = 70 cm. The
distance of the centre of mass from the origin is -

1.  45 cm

2.  50 cm

3.  30 cm

4.  40 cm

Subtopic:  Center of Mass |
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A circular platform is mounted on a frictionless vertical axle. Its radius R = 2m and its moment of intertia about the axle is . It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of  relative to the ground. Time taken by the man to complete one revolution is -

1.  $\frac{3\pi }{2}s$

2.

3.  $\frac{\pi }{2}s$

4.

Subtopic:  Angular Momentum |
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The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through -

1.  C

2.  D

3.  A

4.  B

Subtopic:  Moment of Inertia |
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A uniform rod of length l and mass M is free to rotate in a vertical plane about A. The rod, initially in the horizontal position, is released. The initial angular acceleration of the rod is: (Moment of inertia of the rod about A is$\frac{{\mathrm{Ml}}^{2}}{3}$)

1. $\frac{3g}{2l}$

2. $\frac{2l}{3g}$

3. $\frac{3g}{2{l}^{2}}$

4. $mg\frac{1}{2}$

Subtopic:  Torque |
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The moment of inertia of a uniform circular disc of radius R and mass M about an axis touching the disc at its diameter and normal to the disc is:

1. $M{R}^{2}$

2. $\frac{2}{5}M{R}^{2}$

3. $\frac{3}{2}M{R}^{2}$

4. $\frac{1}{2}M{R}^{2}$

Subtopic:  Moment of Inertia |
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A particle of mass m moves in the XY plane with a velocity v along the straight line AB. If the angular momentum of the particle with respect to origin O is LA when it is at A and LB when it is at B, then:

1. ${L}_{A}>{L}_{B}$

2. ${L}_{A}={L}_{B}$

3. the relationship between LA and LB depends upon the slope of the line AB

4. ${L}_{A}<{L}_{B}$

Subtopic:  Angular Momentum |
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