A force\(- F \hat k\) acts on O, the origin of the coordinate system. The torque at the point (1, -1) will be:

1. $-\mathrm{F}\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)$

2. $\mathrm{F}\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)$

3. $-\mathrm{F}\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}\right)$

4. $\mathrm{F}\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}\right)$

Moment of inertia of an object does not depend upon

A thin uniform circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with an angular velocity $\omega$. Another disc of the same dimensions but of mass \(\frac{1}{4}M\) is placed gently on the first disc co-axially. The angular velocity of the system will be:

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

A flywheel is in the form of a uniform circular disc of radius 1 m and mass 2 kg. The work which must be done on it to increase its frequency of rotation from 5 rev ${\mathrm{s}}^{-1}$ to 10 rev ${\mathrm{s}}^{-1}$ is approximately

1.  1.5 x ${10}^2$ J

2.  3.0 x ${10}^2$ J

3.  1.5 x ${10}^3$ J

4.  3.0 x ${10}^3$ J

A constant torque acting on a uniform circular wheel changes its angular momentum from ${\mathrm{A}}_0$ to $4 {\mathrm{A}}_0$ in 4s. The magnitude of this torque is 

1. $\frac{3{\mathrm{A}}_0}{4}$

2. $4 {\mathrm{A}}_0$

3. ${\mathrm{A}}_0$

4. $12 {\mathrm{A}}_0$

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

A body rolls down an inclined plane without slipping. The fraction of total energy associated with its rotation will be

$1. {\mathrm{K}}^2+ {\mathrm{R}}^2$
$2. \frac{{\mathrm{K}}^2}{{\mathrm{R}}^2}$
$3. \frac{{\mathrm{K}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$
$4. \frac{{\mathrm{R}}^2}{{\mathrm{K}}^2+ {\mathrm{R}}^2}$

Where k is radius of gyration of the body about an axis passing through centre of mass and R is the radius of the body. 

In the following figure, a weight W is attached to a string wrapped round a solid cylinder of mass M mounted on a frictionless horizontal axle at O.

If the weight starts from rest and falls a distance h, then its speed at that instant is

1. Proportional to \(\text R\)

2. Proportional to \(1 \over R\)

3. Proportional to \(1 \over R^2\)

4. Independent of \(\text R\)