A body of mass \(M\) and radius \(R\) is rolling horizontally without slipping with speed \(v.\) It then rolls up a hill to a maximum height \(h.\) If \(h=\frac{5v^{2}}{6g},\) what is the moment of inertia of the body?
1. \(\frac{MR^{2}}{2}\)
2. \(\frac{2MR^{2}}{3}\)
3. \(\frac{3MR^{2}}{4}\)
4. \(\frac{2MR^{2}}{5}\)

A wheel of radius R rolls on the ground with a uniform velocity v. The velocity of topmost point relative to the bottommost point is

1. v

2. 2v

3. v/2

4. zero

If the net external forces acting on the system of particles is zero, then which of the following may vary ?

1. Momentum of the system

2. Velocity of centre of mass

3. Position of centre of mass

4. None of the above

Point masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are placed at the opposite ends of a rigid of length L and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\mathrm{\omega }}_0$ is minimum is given by

$1. \mathrm{x} =\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\mathrm{L}$
$2. \mathrm{x} = \frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}\mathrm{L}$
$3. \mathrm{x} = \frac{{\mathrm{m}}_2\mathrm{L}}{{\mathrm{m}}_1+ {\mathrm{m}}_2}$
$4. \mathrm{x} = \frac{{\mathrm{m}}_1\mathrm{L}}{{\mathrm{m}}_1+ {\mathrm{m}}_2}$

A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force '$\overrightarrow{\mathrm{F}}$' is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C

1.  \(\frac{4}{3} l\)

2.  \(l\)

3.  \(\frac{2}{3} l\)

4.  \(\frac{3}{2} l\)

A cylinder of mass 'M' is suspended by two strings wrapped around it as shown. The acceleration 'a' and the tension T when the cylinder falls and the string unwinds itself are, respectively,
             

1.  $\mathrm{a} = \mathrm{g}, \mathrm{T} = \frac{\mathrm{Mg}}{2}$

2.  $\mathrm{a} = \frac{g}{2}, \mathrm{T} = \frac{\mathrm{Mg}}{2}$

3.  $\mathrm{a} = \frac{g}{3}, \mathrm{T} = \frac{\mathrm{Mg}}{3}$

4.  $\mathrm{a} = \frac{2g}{3}, \mathrm{T} = \frac{\mathrm{Mg}}{6}$

An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver ?

1. 350 N-m

2. 440 N-m

3. 531 N-m

4. 628 N-m

In an orbital motion, the angular momentum vector is 

1. Along the radius vector

2. Parallel to the linear momentum

3. In the orbital plane

4. Perpendicular to the orbital plane

A sphere is rolling down a plane of inclination $\theta$ to the horizontal.  The acceleration of its centre down the plane is 

1. g sin $\theta$                   

2. less than g sin $\theta$

3. greater than g sin $\theta$

4. zero

A bob of mass m attached to an inextensible string of length l is suspended from vertical support.  The bob rotates in a horizontal circle with an angular speed $\mathrm{\omega }$ $\mathrm{rad}/\mathrm{s}$ about the vertical.  About the point of suspension.