Three-point masses 'm' each, are placed at the vertices of an equilateral triangle of side a. Moment of inertia of the system about axis COD is-

1. $2m{a}^2$

2. $\frac{2}{3}m{a}^2$

3. $\frac{5}{4}m{a}^2$

4. $\frac{7}{4}m{a}^2$

A particle is moving in a circular orbit with constant speed. Select wrong alternate

One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively \(\text{I}_{A}\) and \(\text{I}_{B}\) such that
1. \(\text{I}_{\text{A}}=\text{I}_{\text{B}}\)
2. \(\text{I}_{\text{A}}>\text{I}_{\text{B}}\)
3. \(\text{I}_{\text{A}}<\text{I}_{\text{B}}\)
4. \(\frac{\text{I}_{\text{A}}}{\text{I}_{\text{B}}}=\frac{d_A}{d_B}\)

A couple produces:

1. Purely linear motion

2. Purely rotational motion

3. Linear and rotational motion

4. No motion

The radius of gyration of a uniform rod of length \(L\) about an axis passing through its centre of mass is
1. \(\frac{L}{2 \sqrt{3}}\)
2. \(\frac{L^2}{12}\)
3. \(\frac{L}{\sqrt{3}}\)
4. \(\frac{L}{\sqrt{2}}\)

One circular ring and one circular disc both having the same mass and radius. The ratio of their moments of inertia about the axes passing through their centres and perpendicular to planes will be

1. 1:1

2. 2:1

3. 1:2

4. 4:1

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

A constant torque of 1000 N-m turns a wheel of moment of inertial 200 $\mathrm{kg}-{\mathrm{m}}^2$ about an axis through its centre. Its angular velocity after 3 s is

1. 1 rad/s

2. 5 rad/s

3. 10 rad/s

4. 15 rad/s

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

$1.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\mathrm{L}$
$2.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}\mathrm{L}$
$3.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{{\mathrm{m}}_2\mathrm{L}}{{\mathrm{m}}_1+<mspace\; width="1em"></mspace>{\mathrm{m}}_2}$
$4.<mspace\; width="1em"></mspace>\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{{\mathrm{m}}_1\mathrm{L}}{{\mathrm{m}}_1+<mspace\; width="1em"></mspace>{\mathrm{m}}_2}$