A round disc of the moment of inertia ${\mathrm{l}}_2$ about its axis perpendicular to its plane and passing through its center is placed over another disc of the moment of inertial ${\mathrm{l}}_1$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of discs is

1.  $\frac{{\mathrm{l}}_2\mathrm{\omega }}{{\mathrm{l}}_1 + {\mathrm{l}}_2}$

2.  $\mathrm{\omega }$

3.  $\frac{{\mathrm{l}}_1\mathrm{\omega }}{{\mathrm{l}}_1 + {\mathrm{l}}_2}$

4.  $\frac{\left({\mathrm{l}}_1 + {\mathrm{l}}_2\right)\mathrm{\omega }}{{\mathrm{l}}_1}$

Which of the following will not be affected if the radius of the sphere is increased while keeping mass constant?

A particle of mass 5 g is moving with a uniform speed of $3\sqrt{2}$cm/s in the x-y plane along the line y = $2\sqrt{5}$cm. The magnitude of its angular momentum about the origin in $\mathrm{g}-{\mathrm{cm}}^2/\mathrm{s}$ is

1.  0

2.  30

3.  $30\sqrt{2}$

4.  $30\sqrt{10}$

A mass m is supported by a massless string wound round a uniform cylinder of mass M and radius R. On releasing the mass from rest, it will fall with an acceleration assuming no slipping-

1.  $\frac{\mathrm{g}}{3}$

2.  $\frac{\mathrm{g}}{2}$

3.  $\frac{2\mathrm{g}}{3}$

4.  g

A closed tube partly filled with water lies in a horizontal plane. If the tube is rotated about perpendicular bisector, the moment of inertia of the system

1. increases

2. decreases

3. remains constant

4. depends on sense of rotation

A disc is rotating With an angular speed of m. If a child sits on it, which of the following is conserved? 

1.  kinetic energy 

2.  potential energy 

3.  linear momentum 

4.  angular momentum

A particle moves in a force field given by $\overrightarrow{\mathrm{F}} = \hat{\mathrm{r}} \mathrm{F} \left(\mathrm{r}\right)$, where r is the unit vector along with the position vector $\overrightarrow{\mathrm{r}}$, then which is true? 

1.  The torque acting on the particle is not zero 

2.  The torque acting on the particle produces an angular acceleration in it 

3.  The angular momentum of the particle is conserved 

4.  The angular momentum of the particle increases.

Four masses are fixed on a massless rod as shown in the figure. The moment of inertia about the axis P is about

1.  2 kg-${\mathrm{meter}}^2$

2.  1 kg-${\mathrm{meter}}^2$

3.  0.5 kg-${\mathrm{meter}}^2$

4.  0.3 kg-${\mathrm{meter}}^2$

A circular plate of diameter d is kept in contact with a square plate of an edge as shown in the figure. The density of the material and the thickness are the same everywhere. The center of mass of the composite system will be

A small disc of radius 2 cm is cut from a disc of radius 6 cm. If the distance between their centres is 3.2 cm, what is the shift in the centre of mass of the disc?

1.  0.4 cm

2.  2.4 cm

3.  1.8 cm

4.  1.2 cm