Velocity: Bottom Most Point

Q. Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is $\frac{\sqrt{x}}{2}$. Then, the value of $x$ is _____.

Q. A uniform sphere of mass $500\text{g}$ rolls without slipping on a plane horizontal surface with its centre moving at a speed of $5.00\text{cm/s}$. Its kinetic energy is

1. $8.75\times {10}^{-4}\text{J}$
2. $8.75\times {10}^{-3}\text{J}$
3. $6.25\times {10}^{-4}\text{J}$
4. $1.13\times {10}^{-3}\text{J}$

Q. A solid sphere is released from rest from the top of an inclined plane of inclination $\theta$ and length $l$. If the sphere rolls without slipping, what will be its speed when it reaches the bottom?

1. $\sqrt{\frac{10}{7}gl\sin \theta }$
2. $\sqrt{\frac{7}{10}gl\sin \theta }$
3. $\sqrt{\frac{3}{7}gl\sin \theta }$
4. $\sqrt{\frac{7}{3}gl\sin \theta }$

Q. A sphere of mass $m$ hold at a height $2R$ between a wedge of same mass $m$ and a rigid wall, is released from rest. Assuming that all the surfaces are frictionless. Find the speed of sphere when it hits the ground.

1. $\sqrt{2gR}\sin \alpha$
2. $\sqrt{2gR}\cos \alpha$
3. $\sqrt{gR}\sin \alpha$
4. $\sqrt{gR}\cos \alpha$

Q. A disc of radius $5\text{cm}$ rolls on a horizontal surface with linear velocity $v=1\hat{i}\text{m/s}$ and angular velocity $50(-\hat{k})\text{rad/sec}$ . Height of particle from ground on rim of disc which has velocity in vertical direction is (in $\text{cm}$) -

Q. A sphere of mass $m$ is rolling on a frictionless surface (as shown in the figure) with a translation velocity $v\text{m/s}$. The sphere will climb on the inclined plane if $v$ is

1. $\ge \sqrt{\frac{5}{7}gh}$
2. $\ge \sqrt{2gh}$
3. $\ge \sqrt{\frac{10}{7}gh}$
4. $\ge \sqrt{gh}$

Q. A tangential force $10\text{N}$ acts at the top of a thin spherical shell of mass $2\text{kg}$. Find the acceleration of the shell if it rolls without slipping.

1. $3{\text{m/s}}^2$
2. $4{\text{m/s}}^2$
3. $6{\text{m/s}}^2$
4. $8{\text{m/s}}^2$

Q. Two plates 1 and 2 move with velocities $-v$ and $2v$ respectively. If the sphere does not slide relative to the plates, assuming the masses of each body as $m$, find the kinetic energy of the system (plate+sphere).

1. $\frac{125}{40}m{v}^2$
2. $\frac{123}{40}m{v}^2$
3. $\frac{23}{90}m{v}^2$
4. $\frac{23}{40}m{v}^2$

Q. A smooth sphere of radius $R$ is made to translate in a straight line with a constant acceleration $a=g$. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. The speed of particle with respect to the sphere as a function of angle $\theta$ as it slides down is :
Here $g$ is acceleration due to gravity.

1. $\frac{\sqrt{Rg(\sin \theta +\cos \theta )}}{2}$
2. $\sqrt{Rg(1+\cos \theta -\sin \theta )}$
3. $\sqrt{4Rg\sin \theta }$
4. $\sqrt{\left(2Rg\right(1+\sin \theta -\cos \theta )}$

Q.

What is the formula of rotational motion?

Q. A small block is connected to one end of a massless spring of unstretched length \(4.9~\text{m}\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2~\text{m}\) and then released. It then executes SHM with an angular frequency of \(\omega = \left(\dfrac{\pi}{3}\right) \text{rad s}^{-1}\). Simultaneoulsy, at \(t = 0\), a small pebble is projected with a speed \(v\) from the point \(P\) at an angle of \(45^{\circ}\) as shown in the figure. The point \(P\) is at a horizontal distance of \(10~\text{cm}\) from \(O\). If this pebble hits the block at \(t =1~\text{s}\), the value of \(v\) is

10m 

Q. A body rolls down an inclined plane. If its kinetic energy of rotation is $40$% of its kinetic energy of translation motion, then the body is

1. Hollow cylinder
2. Ring
3. Hollow sphere
4. Solid sphere
5. Solid disc

Q. A disc is rolling on an inclined plane without slipping then what fraction of its total energy will be in form of rotational kinetic energy :-

1. $1:3$
2. $1:2$
3. $2:7$
4. $2:5$

Q. A smooth sphere of radius $R$ is set into translatory motion in a straight line with a constant acceleration $a=g{\text{m/s}}^2$. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. If the particle slides down, then the speed of the particle with respect to the sphere when $\theta ={45}^{\circ }$ is :(Here $g=10{\text{m/s}}^2$ is acceleration due to gravity and Radius, $R=0.8\text{m}$).

Q. A car whose radius of tire is 30 cm is travelling at a speed of 54 km/h on a straight road. Find angular speed of tire(assume there is no slipping between tire & road).

1. 20 rad/sec
2. 180 rad/sec
3. 50 rad/sec
4. 10 rad/sec

Q. A heavy solid sphere is thrown on a horizontal rough surface with initial velocity $u$ without rolling. What will be its speed, when it starts pure rolling motion?

1. $\frac{2u}{7}$
2. $\frac{3u}{5}$
3. $\frac{2u}{5}$
4. $\frac{5u}{7}$

Q. Find ratio of rotational and translation kinetic energy of a rolling sphere.

1. $\frac{2}{5}$
2. $\frac{2}{3}$
3. $\frac{1}{6}$
4. $\frac{1}{2}$

Q. What is the kinetic energy of a ball of mass 500 g and moving with a speed of 2 $m{s}^{-1}$ ?

1. 4 J
2. 1 J
3. 41 N
4. 4 ergs

Q.

Uniform metre rule of weight $2.0N$ is pivoted at $60cm$ mark. A $4.0N$ weight is suspended from one end, causing the rule to rotate about the pivot.

At the instant when rule is horizontal, what is the resultant turning moment about the pivot?

1. zero
2. $1.4Nm$
3. $1.6Nm$
4. $1.8Nm$

Q. A tram moves with a velocity $v$. A particle moving horizontally with a speed $u$ enters through corner $B$ perpendicular to $v$. this particle strikes the diagonally opposite corner $A$. If the dimensions of tram are $16\times 2.4\times 3.2m^3$, the value of $v$ is

1. $3m{s}^{-1}$
2. $15m{s}^{-1}$
3. $30m{s}^{-1}$
4. $20m{s}^{-1}$

Q. A small sphere rolls down without slipping from the top of a track in a vertical plane. The track has an elevated section and a horizontal part. The horizontal part is $1m$ above the ground and the top of the track is $2.4m$ above the ground. The horizontal distance traveled by the sphere from the edge of the track where it lands on the ground is nearly :

1. $1m$
2. $2m$
3. $3m$
4. $4m$

Q. The total kinetic energy of rolling solid sphere having translational velocity $\nu$ is

1. $\frac{7}{10}$m${\nu }^2$
2. $\frac{1}{2}$m${\nu }^2$
3. $\frac{2}{5}$m${\nu }^2$
4. $\frac{10}{7}$m${\nu }^2$

Q. A car whose radius of tire is 30 cm is travelling at a speed of 54 km/h on a straight road. Find angular speed of tire(assume there is no slipping between tire & road).

1. 20 rad/sec
2. 180 rad/sec
3. 50 rad/sec
4. 10 rad/sec

Q. A loop and a disc roll without slipping with same linear velocity $v$. The mass of the loop and the disc is same. If the total kinetic energy of the loop is $8J$, find the kinetic energy of the disc (in $J$)

Q. A gymnast on ice folds his arms while rotation. Which of the following statements is true?

1. Speed of rotation increases as friction decreases
2. Rotational kinetic energy decreases because moment of inertia decreases
3. Rotational kinetic energy increases as angular momentum remains conserved
4. None of the above

Q. A wheel takes m rotations to travel a certain distance while another wheel whose diameter is four times bigger than it takes n rotations to travel the same distance. The relation between

1. m $=$ 4n
2. m $=$ 2n
3. 2m $=$ n
4. 4m $=$ n

Q. Two plates 1 and 2 move with velocities $-v$ and $2v$ respectively. If the sphere does not slide relative to the plates, assuming the masses of each body as $m$, find the kinetic energy of the system (plate+sphere).

1. $\frac{125}{40}m{v}^2$
2. $\frac{123}{40}m{v}^2$
3. $\frac{23}{90}m{v}^2$
4. $\frac{23}{40}m{v}^2$

Q. If a solid sphere of mass M and radius R is rolling perfectly on a rough horizontal surface, what is the percentage of the rotational kinetic energy

1. 50%
2. 71.5%
3. 28.57%
4. 10%

Q.

Consider a wheel purely rolling on a rough horizontal surface with constant velocity v. Radius of the wheel is R and C is the center of wheel. M is top-most point, P is bottom-most point and N is in level with C at any time. Match the columns for this instant of time.

1. (p-w, x); (q-x, y); (r-w, x); (s-w, x)

2. (p-w) (q-x, y) (r-w, y) (s-w, z)

3. (p-y, z) (q-y, z) (r-w, x) (s-y)

4. (p(w, x) (q-w, x) (r-w, x) (s-w, x)

Q. A solid sphere of radius ${}^{\prime }{a}^{\prime }$ and mass ${}^{\prime }{m}^{\prime }$ rolls along the horizontal plane with constant speed $v_o$. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping, how far up the iclined plane, sphere will travel (along the incline)?

1. $\frac{2}{5}\frac{v_o^2}{g\sin \theta }$
2. $\frac{10}{7}\frac{v_o^2}{g\sin \theta }$
3. $\frac{1}{5}\frac{v_o^2}{g\sin \theta }$
4. $\frac{7}{10}\frac{v_o^2}{g\sin \theta }$