Velocity of a Point

Q. A wheel of radius $R$ as shown in figure, is set in pure rolling. Point $P$ represents the point on the wheel, in contact with a point $Q$ on the ground. Which of the following statement(s) is/are correct?

1. There is no relative motion at the point of contact between wheel and ground.
2. Wheel will perform pure rolling if velocity of point $P$ is zero.
3. $V_{\text{cm}}=\omega R$
4. None of these

Q. If the ring performs combined rotational and translational motion on a frictionless horizontal surface as shown in figure, find the magnitude of the velocity at a point $P$ on the ring.

1. $2\sqrt{19}\text{m/s}$
2. $\sqrt{38}\text{m/s}$
3. $\sqrt{29}\text{m/s}$
4. $\sqrt{19}\text{m/s}$

Q. A ring of radius $1\text{m}$ is performing combined translational and rotational motion on a frictionless horizontal surface with an angular velocity of $3\text{rad/s}$ as shown in figure. Find out velocity of the point $A$ in $\text{m/s}$, if the speed of the lowest point $V_P$ is $1\text{m/s}$.

1. $4\hat{i}+3\hat{j}$
2. $4\hat{i}-3\hat{j}$
3. $-3\hat{i}+4\hat{j}$
4. $-4\hat{i}-3\hat{j}$

Q. A disc of radius $r$ is performing rotational and translational motion as shown in figure.



Choose the correct option.

1. Velocity of point $A$ w.r.t point $B$ will always remain constant.
2. Speed of point $A$ w.r.t point $B$ will always remain constant.
3. Magnitude of relative velocity of point $A$ w.r.t point $B$ is $2\omega |{\overrightarrow{r}}_{AB}|\text{m/s}$.
4. None of these.

Q. Consider a bicycle tyre rolling without slipping on a smooth horizontal surface with a linear velocity $v_0$ as shown in figure. Then, which of the following statement is incorrect?

1. Speed of point $A$ is zero
2. Speed of point $B,C$ and $D$ are equal to $v_0$
3. Speed of point $B>$ speed of point $O$
4. Speed of point $C=2v_0$

Q. A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, $A$ is the point of contact, $B$ is the centre of the sphere and $C$ is its topmost point. Then,

1. ${\overrightarrow{V}}_C-{\overrightarrow{V}}_A=2({\overrightarrow{V}}_B-{\overrightarrow{V}}_C)$
2. ${\overrightarrow{V}}_C-{\overrightarrow{V}}_B={\overrightarrow{V}}_B-{\overrightarrow{V}}_A$
3. $|{\overrightarrow{V}}_C-{\overrightarrow{V}}_A|=2|{\overrightarrow{V}}_B-{\overrightarrow{V}}_C|$
4. $|{\overrightarrow{V}}_C-{\overrightarrow{V}}_A|=4|{\overrightarrow{V}}_B|$

Q.

A slender uniform rod of mass M and length L is pivoted at one end so that it can rotate in a vertical plane (see the figure).

There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical, is

1. $\frac{2g}{3l}sin\theta$

2. $\frac{3g}{2l}cos\theta$

3. $\frac{2g}{3l}cos\theta$

4. $\frac{3g}{2l}sin\theta$

Q. A solid sphere of radius $r=2\text{m}$ is under a combination of rotational and translational motion as shown in figure.


If acceleration of COM $\left(O\right)$ is $a_o=2{\text{m/s}}^2$ and horizontal component of acceleration at point $A$ is $a_A=6{\text{m/s}}^2$, then find angular acceleration $\alpha$.

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

Q. A system of identical cylinders and plates is shown in the figure. All the cylinders are identical and there is no slipping at any contact. The velocity of lower and upper plates are $V$ and $2V$ respectively. The ratio of angular speeds of the upper cylinders to lower cylinders is

1. $1/3$
2. $3$
3. $1$
4. None of these

Q. A disc is rolling (without slipping) on a horizontal surface. $C$ is its centre and $Q$ and $P$ are two points equidistant from $C$ as shown in figure. Let $v_P,v_Q$ and $v_C$ be the magnitude of velocities of points $P,Q$ and $C$ respectively, then

1. $v_Q>v_C>v_P$
   
2. $v_Q<v_C<v_P$
   
3. $v_Q=v_P,v_C=\frac{1}{2}{v}_P$
   
   
4. $v_Q=v_C=v_P$
   

Q. Two identical discs of the same radius R are rotating about their axes in opposite directions with the same constant angular speed $\omega .$ The discs are in the same horizontal plane. At time t = 0, the points P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is $v_r$. In one time period (T) of rotation of the discs, $v_r$ as a function of time is best represented by

1. 
2. 
3. 
4. 

Q. If the ring performs combined rotational and translational motion on a frictionless horizontal surface as shown in figure, find the magnitude of the velocity at a point $P$ on the ring.

1. $2\sqrt{19}\text{m/s}$
2. $\sqrt{38}\text{m/s}$
3. $\sqrt{29}\text{m/s}$
4. $\sqrt{19}\text{m/s}$

Q. A gramophone record is revolving with angular velocity $\omega$. A coin is placed at a distance $r$ from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if

Q.

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $\frac{\omega }{2}$. The ring and disc are separated by frictionless ball bearings. The system is in the x - z place. The point P on the inner disc is at distance R from the origin O, where OP makes an angled of 30${}^{\circ }$ with the horizontal. Then with respect to the horizontal surface,

1. The point O has a linear velocity $3R\omega \hat{i}$

2. The point P has a linear velocity $\frac{11}{4}R\omega \hat{i}-\frac{\sqrt{3}}{4}R\omega \hat{k}$

3. The point P has a linear velocity $\frac{13}{4}R\omega \hat{i}-\frac{\sqrt{3}}{4}R\omega \hat{k}$

4. The point P has a linear velocity $(3-\frac{\sqrt{3}}{4})R\omega \hat{i}+\frac{1}{4}R\omega \hat{k}$

Q. A system of identical cylinders and plates is shown in the figure. All the cylinders are identical and there is no slipping at any contact. The velocity of lower and upper plates are $V$ and $2V$ respectively. The ratio of angular speeds of the upper cylinders to lower cylinders is

1. $1/3$
2. $3$
3. $1$
4. None of these

Q.

On the cylinder of radius R in the figure, which is rolling with velocity of centre of mass $v=\frac{\omega R}{4}$ towards right ($\omega$ is clockwise), find the velocities (magnitude) at points A, B, C, D.

1. $5v,\sqrt{17}v,3v,\sqrt{17}v$

2. $5v,\sqrt{2}v,3v,\sqrt{2}v$

3. $2v,\sqrt{2}v,0,\sqrt{2}v$

4. $2v,\sqrt{17}v,0,\sqrt{17}v$

Q. The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius 2R rotating anticlockwise with angular speed $\frac{\omega }{2}.$ The ring and the disc are separated by frictionless ball bearings. The system is in the xz-plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of ${30}^{\circ }$ with this horizontal, where O is the centre of the disc. Then with respect to this horizontal surface

1. The point O has linear velocity $3R\omega \hat{i}$
2. The point P has linear velocity $(\frac{11}{4}R\omega \hat{i}+\frac{\sqrt{3}}{4}R\omega \hat{k})$
3. The point P has linear velocity $(\frac{13}{4}R\omega \hat{i}-\frac{\sqrt{3}}{4}R\omega \hat{k})$
4. The point P has linear velocity $((3-\frac{\sqrt{3}}{4})R\omega \hat{i}+\frac{1}{4}R\omega \hat{k})$

Q. The velocity of centre of mass of a rolling disc is $v$. Considering sufficient friction to prevent any slipping, the magnitude of velocity at the highest point on the disc is

1. $\sqrt{2}v$
2. $v$
3. $2v$
4. $\frac{v}{\sqrt{2}}$

Q.

A cylinder of radius R is rolling with velocity of centre of mass $v=\frac{\omega R}{2}$ towards right. Keeping the centre of the cylinder as the origin, Find the coordinates of the point with zero net velocity.

1. $\frac{R}{2},0$

2. $0,\frac{-R}{2}$

3. $\frac{R}{\sqrt{2}},\frac{R}{\sqrt{2}}$

4. $0,\frac{R}{\sqrt{2}}$