The minimum coefficient of friction for a solid sphere to roll without slipping on an inclined plane of inclination 45$°$ is

1.  2/7

2.  1/3

3.  1/2

4.  2/5

A wheel is rolling along a horizontal ground with a speed o 10 m/s. The magnitude of the velocity of the points at the extremities of the horizontal diameter of the wheel is equal to

1. $2\sqrt{10}<mspace\; width="1em"></mspace><mspace\; width="1em"></mspace>m/s$

2. 10 m/s

3. $\sqrt{10}<mspace\; width="1em"></mspace>m/s$

4. $10\sqrt{2}<mspace\; width="1em"></mspace>m/s$

$\mathrm{A}<mspace\; width="1em"></mspace>\mathrm{body}<mspace\; width="1em"></mspace>\mathrm{of}<mspace\; width="1em"></mspace>\mathrm{mass}<mspace\; width="1em"></mspace>\mathrm{m}<mspace\; width="1em"></mspace>\mathrm{is}<mspace\; width="1em"></mspace>\mathrm{projected}<mspace\; width="1em"></mspace>\mathrm{at}<mspace\; width="1em"></mspace>\mathrm{a}<mspace\; width="1em"></mspace>\mathrm{velocity}<mspace\; width="1em"></mspace>{\mathrm{v}}_0$
$\mathrm{with}<mspace\; width="1em"></mspace>\mathrm{an}<mspace\; width="1em"></mspace>\mathrm{angle}<mspace\; width="1em"></mspace>\mathrm{of}<mspace\; width="1em"></mspace>\mathrm{projection}<mspace\; width="1em"></mspace>\mathrm{\theta }.<mspace\; width="1em"></mspace>\mathrm{Angular}<mspace\; width="1em"></mspace>\mathrm{momentum}$
$\mathrm{L}<mspace\; width="1em"></mspace>\mathrm{of}<mspace\; width="1em"></mspace>\mathrm{body}<mspace\; width="1em"></mspace>\mathrm{after}<mspace\; width="1em"></mspace>\mathrm{time}<mspace\; width="1em"></mspace>\mathrm{t}<mspace\; width="1em"></mspace>\mathrm{w}.\mathrm{r}.\mathrm{to}<mspace\; width="1em"></mspace>\mathrm{origin}<mspace\; width="1em"></mspace>\mathrm{is}<mspace\; width="1em"></mspace>\mathrm{proportional}<mspace\; width="1em"></mspace>\mathrm{to}:$
$1.<mspace\; width="1em"></mspace>\mathrm{L}\propto \mathrm{t}<mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace>2.<mspace\; width="1em"></mspace>\mathrm{L}\propto {\mathrm{t}}^2$
$3.<mspace\; width="1em"></mspace>\mathrm{L}\propto {\mathrm{t}}^0<mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace><mspace\; width="1em"></mspace>4.<mspace\; width="1em"></mspace>\mathrm{L}\propto \sqrt{\mathrm{t}}$

A mass M is moving with constant velocity parallel to the X-axis. Its angular momentum with respect to origin is:

1.  Zero

2.  Remains constant

3.  goes on increasing

4.  goes on decreasing

If I is the moment of inertia of a solid sphere about an axis parallel to diameter of a shpere and at a distance x from it. Which of the following graph represents the variation of I with X?

1.		2.
3.		4.

A particle is rotating under the central force, in which one of the following quantities is constant:

1.  torque

2.  angular momentum

3.  linear momentum

4.  none of these

A small object of uniform density rolls up a curved surface with an initial velocity \(v.\) It is reached upto a maximum height of \(\frac{3v^{2}}{4g}\) with respect to the initial position. The object is:

1. ring
2. solid sphere
3. hollow sphere
4. disc

A point is at (x, y, z). What is its moment of inertia about x-axis:

1.  $\mathrm{m}\left({\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\right)$

2.  $\mathrm{m}\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)$

3.  $\mathrm{m}\left({\mathrm{y}}^2+{\mathrm{z}}^2\right)$

4.  $\mathrm{m}\left({\mathrm{x}}^2+{\mathrm{z}}^2\right)$

The disc is placed on the rough horizontal surface given only angular velocity ${\omega }_0$. The velocity of the centre of mass when it starts pure rolling is:

1.  $\frac{{\omega }_{0}R}{2}$

2.  $\frac{{\omega }_{0}R}{3}$

3.  $\frac{{\omega }_{0}R}{4}$

4.  $\frac{{\omega }_{0}R}{6}$

A force 2N is along line y = 3x + 4. The torque about the origin is:

1.  $\frac{5}{\sqrt{10}}\mathrm{Nm}$

2.  $\frac{7}{\sqrt{10}}\mathrm{Nm}$

3.  $\frac{6}{\sqrt{10}}\mathrm{Nm}$

4.  $\frac{8}{\sqrt{10}}\mathrm{Nm}$