A mass m moves in a circle on a smooth horizontal plane with velocity v0 at a radius R0. The mass is attached to a string that passes through a smooth hole in the plane as shown.

The tension in the string is increased gradually and finally m moves in a circle of radius$\frac{{R}_{0}}{2}$. The final value of the kinetic energy is:
1. $m{v}_{0}^{2}$
2. $\frac{1}{4}m{v}_{0}^{2}$
3. 2$m{v}_{0}^{2}$
4. $\frac{1}{2}m{v}_{0}^{2}$

Subtopic:  Angular Momentum |
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A force $\stackrel{\to }{F}=\alpha \stackrel{^}{i}+3\stackrel{^}{j}+6\stackrel{^}{k}$ is acting at a point $\stackrel{\to }{r}=2\stackrel{^}{i}-6\stackrel{^}{j}-12\stackrel{^}{k}$. The value of $\alpha$ for which angular momentum about the origin is conserved is:

1. -1

2. 2

3. zero

4. 1

Subtopic:  Angular Momentum |
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When a mass is rotating in a plane about a fixed point, its angular momentum is directed along:

1. a line perpendicular to the plane of rotation

2. the line making an angle of 45o to the plane of rotation

4. the tangent to the orbit

Subtopic:  Angular Momentum |
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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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A circular disk of a moment of inertia It is rotating in a horizontal plane, about its symmetric axis, with a constant angular speed ${\mathrm{\omega }}_{\mathrm{i}}$. Another disk of a moment of inertia Ib is dropped coaxially onto the rotating disk. Initially, the second disk has zero angular speed. Eventually, both the disks rotate with a constant angular speed ${\mathrm{\omega }}_{\mathrm{f}}$. The energy lost by the initially rotating disc due to friction is:

1. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}^{2}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^{2}$

2. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{t}}^{2}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^{2}$

3. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}-{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^{2}$

4. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega }}_{\mathrm{i}}^{2}$

Subtopic:  Angular Momentum |
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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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A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:

1. $\frac{\omega \left(M-2m\right)}{M+2m}$

2. $\frac{\omega M}{M+2m}$

3. $\frac{\omega \left(M+2m\right)}{M}$

4. $\frac{\omega M}{M+m}$

Subtopic:  Angular Momentum |
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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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