A small mass attached to a string rotates on a frictionless table top as shown. If the tension on the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of \(2,\) the kinetic energy of the mass will

A couple produces:

1. Purely linear motion

2. Purely rotational motion

3. Linear and rotational motion

4. No motion

A solid sphere is rotating about a diameter at an angular velocity \(w.\) If it cools so that its radius reduces to$\frac{\mathrm{}}{}$\(\frac1n\) of its  original value, its angular velocity becomes
1. \(\frac wn\) 
2. \(\frac{w}{{n}^2}\)
3. \(nw\)
4. \(n^2w\)

A horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass 'm' is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period

1. Decreases continuously

2. Decreases initially and increases again

3. Remains unaltered

4. Increases continuously

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

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

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

When a torque acting upon a system is zero, then which of the following will be constant 

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D${}_1$ has 2 kg mass and 0.2 m radius and 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 is

1. 60

2. 100

3. 120

4. 40