A solid cylinder of mass \(3\) kg is rolling on a horizontal surface with a velocity of \(4\) ms^-1. It collides with a horizontal spring of force constant \(200\) Nm^-1. The maximum compression produced in the spring will be:
1. \(0.5\) m
2. \(0.6\) m
3. \(0.7\) m
4. \(0.2\) m

A wheel of radius R rolls without slipping on the ground with a uniform velocity v. The relative acceleration of the topmost point of the wheel with respect to the bottommost point is:

1. $\frac{{\mathrm{v}}^2}{\mathrm{R}}$

2. $\frac{2{\mathrm{v}}^2}{\mathrm{R}}$

3. $\frac{{\mathrm{v}}^2}{2\mathrm{R}}$

4. $\frac{4{\mathrm{v}}^2}{\mathrm{R}}$

The ratio of the acceleration for a solid sphere (mass \(m\) and radius \(R\)) rolling down an incline of angle \(\theta\) without slipping and slipping down the incline without rolling is:
1. \(5:7\)
2. \(2:3\)
3. \(2:5\)
4. \(7:5\) 

A disc and a solid sphere of the same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

1. Sphere

2. Both reach at the same time

3. Depends on their masses

4. Disc

A solid sphere is in rolling motion. In rolling motion, a body possesses translational kinetic energy (K_t) as well as rotational kinetic energy (K_r) simultaneously. The ratio K_t : (K_t + K_r) for the sphere will be:

1. 7:10

2. 5:7

3. 10:7

4. 2:5

A hollow cylinder with an inner radius of R, an outer radius of 2R, and a mass M is rolling without slipping at the speed of its centre v. Its kinetic energy will be:
  

1. $\frac{11}{16}M{v}^2$

2. $\frac{7}{4}M{v}^2$

3. $\frac{13}{16}M{v}^2$

4. None of these

A solid cylinder of mass 2 kg and radius 50 cm rolls up an inclined plane of angle of inclination ${30}^{°}$. The centre of mass of the cylinder has a speed of 4 m/s. The distance travelled by the cylinder on the inclined surface will be $[take$ $g=$ $10$ $m/s^2]\; :$

1. 2.2 m

2. 1.6 m

3. 1.2 m

4. 2.4 m

A ring and a disc with the same moment of inertia roll along a plane surface at the same speed. If ${\mathrm{E}}_{\mathrm{r}}$ be the rotational kinetic energy of the ring and ${\mathrm{E}}_{\mathrm{d}}$ be that of the disc, then:

1. ${\mathrm{E}}_{\mathrm{r}}$>${\mathrm{E}}_{\mathrm{d}}$

2. ${\mathrm{E}}_{\mathrm{r}}$<${\mathrm{E}}_{\mathrm{d}}$

3. ${\mathrm{E}}_{\mathrm{r}}$=${\mathrm{E}}_{\mathrm{d}}$

4. The relation depends upon the radii of the ring and disc

A solid sphere is rolling on a frictionless surface, as shown in the figure with a translational velocity of v m/s. If a sphere climbs up to a height h, then the value of v would be:
                 

1. $\ge \sqrt{\frac{10}{7}\mathrm{gh}}$                                                   

2. $\ge \sqrt{2\mathrm{gh}}$

3. $2\mathrm{gh}$                                                             

4. $\frac{10}{7}\mathrm{gh}$