Three-point masses each of mass \(m,\) are placed at the vertices of an equilateral triangle of side \(a.\) The moment of inertia of the system through a mass \(m\) at \(O\) and lying in the plane of \(COD\) and perpendicular to \(OA\) is:

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

1. Its linear momentum is conserved

2. Its angular momentum is conserved

3. It is moving with variable velocity

4. It is moving with variable acceleration

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 I_A and I_B such that

1. ${\mathrm{I}}_{\mathrm{A}}={\mathrm{I}}_{\mathrm{B}}$

2. ${\mathrm{I}}_{\mathrm{A}}>{\mathrm{I}}_{\mathrm{B}}$

3. ${\mathrm{I}}_{\mathrm{A}}<{\mathrm{I}}_{\mathrm{B}}$

4. $\frac{{\mathrm{I}}_{\mathrm{A}}}{{\mathrm{I}}_{\mathrm{B}}}=\frac{d_A}{d_B}$

A couple produces:   

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

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

$A solid sphere rolls down on an inclined plane of inclination \theta .$
$What is the acceleration as the sphere reaches bottom?$
$\mathbf{[}\mathit{O}\mathit{d}\mathit{i}\mathit{s}\mathit{h}\mathit{a}\mathbf{ }\mathit{J}\mathit{E}\mathit{E}\mathbf{ }\mathbf{2002}\mathbf{,}\mathbf{ }\mathbf{03}\mathbf{;}\mathbf{ }\mathit{C}\mathit{B}\mathit{S}\mathit{E}\mathbf{ }\mathit{P}\mathit{M}\mathit{T}\mathbf{ }\mathbf{2014}\mathbf{]}$
$\left(a\right) \frac{5}{7}gSin\theta \left(b\right) \frac{3}{5}gSin\theta$
$\left(c\right) \frac{2}{7}gSin\theta \left(d\right) \frac{2}{5}gSin\theta$

A wheel is at rest. Its angular velocity increases uniformly and becomes \(80\) rad/s after \(5\) s. The total angular displacement is

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has a 2 kg mass, 0.2 m radius, and an initial angular velocity of 50 rad s${}^{-1}$. Disc D${}_2$ has 4 kg mass, 0.1 m radius, and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s${}^{-1}$) of the system will be:

1. 60

2. 100

3. 120

4. 40