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A particle starts from the point (0,8) meter and moves with a uniform velocity of v = 3$\hat{\mathrm{i}}$ m/s. What is the angular momentum of the particle after 5 s about the origin (mass of the particle is 1 kg)?
1. -12 $\hat{\mathrm{k}}$ kg m/s
2. -24 $\hat{\mathrm{k}}$ kg m/s
3. -32 $\hat{\mathrm{k}}$ kg m/s
4. -36 $\hat{\mathrm{k}}$ kg m/s

A ball of mass 1 kg is projected with a velocity of 20 $\sqrt{2}$ m/s from the origin of a xy co-ordinate axis system at an angle 45° with the x-axis (horizontal). The angular momentum [In SI units) of the ball about the point of projection after 2 s of projection is stake g = 10 m/${\mathrm{s}}^2$) (y-axis is taken as vertical)
1. -400 $\hat{\mathrm{k}}$
2. 200 $\hat{\mathrm{i}}$
3. 300 $\hat{\mathrm{j}}$
4. -350 $\hat{\mathrm{j}}$

The moment of inertia of a uniform semicircular wire of mass m and radius r, about an axis passing through its center of mass and perpendicular to its plane, is
1. $\frac{{\mathrm{mr}}^2}{2}$
2. ${\mathrm{mr}}^2$
3. ${\mathrm{mr}}^2\left(1- \frac{4}{{\mathrm{\pi }}^2}\right)$
4. ${\mathrm{mr}}^2\left(1 + \frac{4}{{\mathrm{\pi }}^2}\right)$

Moment of inertia of a uniform circular disc about its diameter is I. Its moment of inertia about an axis parallel to its plane and passing through a point on its rim will be
1. 3I
2. 4I
3. 5I
4. 6I

Four rings each of mass M and radius R are arranged as shown in the figure. The moment of inertia of the system about the axis yy' is
                                    
1. 2${\mathrm{MR}}^2$
2. 3${\mathrm{MR}}^2$
3. 4${\mathrm{MR}}^2$
4. 5${\mathrm{MR}}^2$

A solid body rotates about a fixed axis such that its angular velocity depends on $\mathrm{\theta }$ as $\mathrm{\omega } = {\mathrm{k\theta }}^{-1}$ where k is a positive constant. At t = 0, 0 = 0, then time dependence of is given as
1. $\mathrm{\theta }$ = kt
2. $\mathrm{\theta }$ = 2kt
3. $\mathrm{\theta }$ = $\sqrt{kt}$
4. $\mathrm{\theta }$ = $\sqrt{2kt}$

A uniform disc of mass m and radius R is pivoted at point P and is free to rotate in the vertical plane. The center C of the disc is initially in a horizontal position with P as shown in the figure. If it is released from this position, then its angular acceleration when the line PC is inclined to the horizontal at an angle $\mathrm{\theta }$ is
                                              
1. $\frac{2\mathrm{g} \cos \mathrm{\theta }}{3\mathrm{R}}$
2. $\frac{\mathrm{g} \sin \mathrm{\theta }}{2\mathrm{R}}$
3. $\frac{2\mathrm{g} \sin \mathrm{\theta }}{\mathrm{R}}$
4. $\frac{2\mathrm{g} \sin \mathrm{\theta }}{3\mathrm{R}}$

A rod of length L leans against a smooth vertical wall while its other end is on a smooth floor. The end that leans against the wall moves uniformly vertically downward. Select the correct alternative
                                  
1. The speed of the lower end increases at a constant rate of
2. The speed of the lower end decreases but never becomes zero
3. The speed of the lower end gets smaller and smaller and vanishes when the upper end touches the ground
4. The speed of the lower end remain constant till the upper end touches the ground

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. Angular momentum
4. Rotational kinetic energy

A solid sphere is in a rolling motion. In rolling motion, a body possesses translational kinetic energy (Kt) as well as rotational kinetic energy (${\mathrm{K}}_{\mathrm{r}}$) simultaneously. The ratio ${\mathrm{K}}_{\mathrm{t}}: \left({\mathrm{K}}_{\mathrm{r}} + {\mathrm{K}}_{\mathrm{t}}\right)$ for the sphere is
1. 7:10
2. 5:7
3. 2:5
4. 10:7

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 $\mathrm{\omega }$ about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation
1. ${\mathrm{W}}_{\mathrm{C}} > {\mathrm{W}}_{\mathrm{B}} > {\mathrm{W}}_{\mathrm{A}}$
2. ${\mathrm{W}}_{\mathrm{A}} >{\mathrm{W}}_{\mathrm{B}} > {\mathrm{W}}_{\mathrm{C}}$
3. ${\mathrm{W}}_{\mathrm{A}} > {\mathrm{W}}_{\mathrm{C}} > {\mathrm{W}}_{\mathrm{B}}$
4. ${\mathrm{W}}_{\mathrm{B}} > {\mathrm{W}}_{\mathrm{A}} > {\mathrm{W}}_{\mathrm{C}}$

The moment of the force, $\mathrm{F} = 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - 6\hat{\mathrm{k}}$at (2, 0, -3), about the point (2, -2, -2), is given by
1. $-8\widehat{\mathrm{î}} - 4\hat{j} - 7\hat{\mathrm{k}}$
2. $-4\hat{\mathrm{i}} -\hat{\mathrm{j}} - 8\hat{\mathrm{k}}$
3. $-7\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 8\hat{\mathrm{k}}$
4. $-7\hat{\mathrm{i}} - 8\hat{\mathrm{j}} - 4\hat{\mathrm{k}}$

A rope is wound around a hollow cylinder of mass 3 kg and a radius of 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N?
1. 25 $\mathrm{m}/{\mathrm{s}}^2$
2. 0.25 $\mathrm{m}/{\mathrm{s}}^2$
3. 25 $\mathrm{rad}/{\mathrm{s}}^2$
4. 5 $\mathrm{m}/{\mathrm{s}}^2$

Which of the following statements are correct?
1. The Centre of the mass of a body always coincides with the center of gravity of the body.
2. The Centre of the mass of a body is the point at which the total gravitational torque on the body is zero
3. A couple on a body produces both translational and rotational motion in a body.
4. Mechanical advantage greater than one means that a small effort can be used to lift a large load.

Two discs of the same moment of inertia rotating about their regular axis passing through center and perpendicular to the plane of the disc with angular velocities ${\mathrm{\omega }}_1 \mathrm{and} {\mathrm{\omega }}_2$. They are brought into contact face to face coinciding with the axis of rotation. The expression for loss of energy during this process is
1. $\frac{1}{2}l{\left({\omega }_1 + {\omega }_2\right)}^2$
2. $\frac{1}{4}l{\left({\omega }_1 - {\omega }_2\right)}^2$
3. $l{\left({\omega }_1 - {\omega }_2\right)}^2$
4. $\frac{1}{8}l{\left({\omega }_1 - {\omega }_2\right)}^2$

Two rotating bodies A and B of masses m and 2m with moments of inertia ${\mathrm{l}}_{\mathrm{A}} \mathrm{and} {\mathrm{l}}_{\mathrm{B}} \left({\mathrm{l}}_{\mathrm{A}} > {\mathrm{l}}_{\mathrm{B}}\right)$ have an equal kinetic energy of rotation. If ${\mathrm{L}}_{\mathrm{A}} \mathrm{and} {\mathrm{L}}_{\mathrm{B}}$ be their angular momenta respectively, then
1. ${\mathrm{L}}_{\mathrm{A}} = {\mathrm{L}}_{\mathrm{B}}/2$
2. ${\mathrm{L}}_{\mathrm{A}} = 2{\mathrm{L}}_{\mathrm{B}}$
3. ${\mathrm{L}}_{\mathrm{A}} < {\mathrm{L}}_{\mathrm{B}}$
4. ${\mathrm{L}}_{\mathrm{A}} > {\mathrm{L}}_{\mathrm{B}}$

A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $({\mathrm{E}}_{\mathrm{sphere}}/{\mathrm{E}}_{\mathrm{cylinder}})$ will be
1. 2:3
2. 1:5
3. 1:4
4. 3:1 

A light rod of length 1 has two masses ${\mathrm{m}}_1, \mathrm{and} {\mathrm{m}}_2$ attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the center of mass is
1. $\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{m}}_1 + {\mathrm{m}}_2}{l}^2$
2. $\frac{m_{\mathit{1}}\mathit{ }\mathit{+}\mathit{ }{m}_{\mathit{2}}}{m_{\mathit{1}}{m}_{\mathit{2}}}{l}^{\mathit{2}}$
3. $(m_1 + m_2)l^{\mathit{2}}$
4. $\sqrt{m_{\mathit{1}}{m}_{\mathit{2}}}{l}^2$

From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the center is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the center?
1. $\frac{9{\mathrm{MR}}^2}{32}$
2. $\frac{15{\mathrm{MR}}^2}{32}$
3. $\frac{13{\mathrm{MR}}^2}{32}$
4. $\frac{11{\mathrm{MR}}^2}{32}$

A disk and a sphere of 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?
1. Depends on their masses
2. Disk
3. Sphere
4. Both reach at the same time