Two uniform, thin, identical rods, each of mass M and length l are joined together to form a cross. What will be the moment of inertia of the cross about an axis passing through the point at which the two rods are joined and are perpendicular to the plane of the cross?

1. $\frac{M{l}^2}{12}$

2. $\frac{M{l}^2}{6}$

3. $\frac{M{l}^2}{4}$

4. $\frac{M{l}^2}{3}$

A wheel is rotating about an axis through its centre at \(720\) r.p.m. It is acted upon by a constant torque opposing its motion for \(8\) seconds to bring it to rest finally. The value of torque in N-m is: (given \(I\) = $\frac{24}{\mathrm{\pi }}$ kg ${\mathrm{m}}^2$)
1. \(48\)
2. \(72\)
3. \(96\)
4. \(120\)

A particle of mass m moves with a constant velocity along 3 different paths, DE, OA and BC. Which of the following statements is not correct about its angular momentum about point O?

A bob of mass m attached to an inextensible string of length l is suspended from vertical support.  The bob rotates in a horizontal circle with an angular speed $\mathrm{\omega }$ $\mathrm{rad}/\mathrm{s}$ about the vertical.  About the point of suspension.

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

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:

1.  $\frac{2}{3}\mathrm{\omega }$

2.  $\frac{4}{5}\mathrm{\omega }$

3.  $\frac{3}{4}\mathrm{\omega }$

4.  $\frac{1}{3}\mathrm{\omega }$

A particle of mass m moves in the XY plane with a velocity of V along the straight line AB. If the angular momentum of the particle about the origin O is L_A when it is at A and L_B when it is at B, then:

A wheel with a radius of 20 cm has forces applied to it as shown in the figure. The torque produced by the forces of 4 N at A, 8N at B, 6 N at C, and 9N at D, at the angles indicated, is:
                    
1.  5.4 N-m anticlockwise

2.  1.80 N-m clockwise

3.  2.0 N-m clockwise

4.  3.6 N-m clockwise

If the radius of the earth is suddenly contracted to half of its present value, then the duration of the day will be of:

If the linear density of a rod of length \(3 \text m\) varies as $$\(\lambda= \text{2+x} \), then the position of the center of mass of the rod is at a distance of:
1. \({7 \over 3}m\)
2. \({10 \over 7}m\)
3. \({12\over 7}m\)
4. \({9 \over 7}m\)