Velocity of a Point

Q.

A rod of mass m and length L, lying horizontally, is free to rotate about a vertical axis through its centre. A horizontal force of constant magnitude F acts on the rod at a distance of $\frac{L}{4}$ from the centre. The force is always perpendicular to the rod. Find the angle rotated by the rod during the time l after the motion starts.

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 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 disc rotating about its axis with angular speed ${\omega }_0$ is placed smoothly (without any translational velocity) on a perfectly frictionless table. Select the correct statement(s):

1. Speed of point $A$ is $v_A={\omega }_{0}R$
2. Speed of point $B$ is zero.
3. Speed of point $C$ is $v_C=\frac{{\omega }_{0}R}{2}$
4. Speed of point $C$ is $v_C={\omega }_{0}R$

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

Q. A wheel is rolling without slipping on ground along a straight line. If the axis of the wheel has a linear speed $v$, the instantaneous velocity of point $P$ at an angle $\theta$ from vertical on the rim, relative to the ground will be

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

Q.

A hollow sphere is released from the top of an inclined plane of inclination $\theta$. (a) What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding ? (b) Find the kinetic energy of the ball as it moves down a length l on the incline if the friction coefficient is half the value calculated in part (a).

Q. A wheel performs pure rolling on the ground. Find a point on the periphery of the body which has a velocity equal to the velocity of the centre of mass of the body.

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

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

Q. A disc of radius $R$ is rolling (without slipping) on a frictionless surface as shown in figure. $C$ is it's centre and $Q$ and $P$ are two points equidistant from $C$. Let $v_P,v_Q$ and $v_C$ be the magnitudes of velocities of points $P$, $Q$ and $C$ respectively, then

1. $v_Q>v_P>v_C$
   
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.

A small particle of mass m is given an initial high velocity in the horizontal plane and winds its cord around the fixed vertical shaft of radius a. Motion occurs essentially in horizontal plane (assume gravity free space). The angular velocity of the cord is ${\omega }_0$ when the distance from the particle to the tangency point is $r_0$.

1. The angular velocity of the cord after it has turned through an angle is

2. The angular velocity of the cord after it has turned through an angle is

3. The KE of the particle will increase

4. The KE of the particle will remain same

Q. A disc of radius $R$ is rolling (without slipping) on a frictionless surface as shown in figure. $C$ is it's centre and $Q$ and $P$ are two points equidistant from $C$. Let $v_P,v_Q$ and $v_C$ be the magnitudes of velocities of points $P$, $Q$ and $C$ respectively, then

1. $v_Q>v_P>v_C$
   
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. 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. A disc of radius $r$ has linear velocity $V$ and angular velocity $\omega$ as shown in figure. If point $B$ is at a distance $\frac{r}{2}$ from centre, then find the velocity of point $A$ w.r.t point $B$ $\left({\overrightarrow{V}}_{AB}\right)$.

1. $\frac{\omega r}{3}$
2. $\omega r$
3. ${\omega }^{2}r$
4. $\frac{\omega r}{2}$

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. In the given figure, an $L$-shaped frame is free to rotate in a vertical plane about a horizontal axis passing through a smooth hinge $O$. Each side of the frame has a length $L$ and mass $M$. The frame is allowed to fall, with one side horizontal and the other vertical.


The linear speed of end $A$ just before striking the ground is

1. $\sqrt{\frac{3gL}{10}(5-\sqrt{3})}$
2. $\sqrt{\frac{3gL}{10}(5+\sqrt{3})}$
3. $\sqrt{\frac{3g}{10L}(5-\sqrt{3})}$
4. $\sqrt{\frac{3g}{10L}(5+\sqrt{3})}$
   

Q. A particle of mass $1\text{g}$ executes an oscillatory motion on the concave surface of a spherical dish of radius $2\text{m}$. The particle begins from a point on the dish at a height of $1\text{cm}$ from the horizontal plane and the coefficient of friction is $0.01$. The total distance covered by the particle before it comes to rest is approximately

1. $2.0\text{m}$
2. $10.0\text{m}$
3. $1.0\text{m}$
4. $20.0\text{m}$

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. A wheel rolls without slipping on a level road as shown in figure.


Choose the correct option(s):

1. The speed of point $B$ is $2V$
2. The speed of points $C,A$ and $D$ is equal to $V$
3. The speed of point $A$ is zero and speed of point $C$ is $\sqrt{2}V$
4. The speed of point $O$ is less than the speed of $C$.

Q. A disc with linear velocity $V$ and angular velocity $\omega$ is placed on the ground.
Suppose $a$ and $\alpha$ be the magnitudes of linear and angular acceleration due to friction. Then match the statements in Column $I$ with statements in column $II$


$\begin{array}{cc}ColumnI& ColumnII\\ \left(A\right)\text{When}V=R\omega & \left(p\right)a=R\alpha (a\ne 0)\\ \text{(B)When}V=\frac{R\omega }{2}& \left(q\right)a>R\alpha \\ \left(C\right)\text{When}V=2R\omega & \left(r\right)a<R\alpha \\ & \left(s\right)\text{None of these}\end{array}$

1. $A\to \left(p\right),B\to \left(r\right),C\to \left(r\right)$
2. $A\to \left(q\right),B\to \left(t\right),C\to \left(p\right)$
3. $A\to \left(s\right),B\to \left(r\right),C\to \left(r\right)$
4. $A\to \left(s\right),B\to \left(r\right),C\to \left(s\right)$

Q.

A circular disc of mass 300 gm and radius 20 cm can rotate freely about a vertical axis passing through is center O. A small insect of mass 100 gm is initially at a point A on the rim. The insect initially stationary starts walking from rest along the rim of disc with such a time varying relative velocity that the disc rotates in the opposite direction with a constant angular acceleration = $2\pi rad/s^2$. After some time T, the insect is back at the point A

. What is the time taken by insect to reach the original position?

1. 

2. 

3. 

4. 

Q. A sphere of radius $0.1\text{m}$ and mass $8\pi \text{kg}$ is attached to the lower end of a steel wire of length $5.0\text{m}$ and diameter ${10}^{-3}\text{m}$. The wire is suspended from $5.22\text{m}$ high ceiling of a room. When the sphere is made to swing as a simple pendulum, it just grazes the floor at its lowest point. Calculate the velocity of the sphere at the lowest position. Young's modulus of steel is $1.994\times {10}^{11}{\text{N/m}}^2$.

1. $2.2\text{m/s}$
2. $4.4\text{m/s}$
3. $8.8\text{m/s}$
4. $12.2\text{m/s}$

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

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 bobbin is pushed along on a rough stationary horizontal surface as shown in the figure. The board is kept horizontal and there is no slipping at any contact points. Find the distance moved by the board when distance moved by the axis of the bobbin is ℓ.

1. ℓr

2. ℓR

3. 

4. 

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

Q. A sphere is moving towards the positive $x$-axis with a velocity $v_c$ and rotates clockwise with angular speed $\omega$ shown in the figure such that $v_c>\omega r$. The instantaneous axis of rotation will be

1. on point $P$
2. on point $P^{{}^{\prime }}$
3. inside the sphere
4. outside the sphere

Q. As shown in figure, a wheel $(r=1\text{m})$ is performing combined translational and rotational motion with angular velocity $2\text{rad/s}$ and the velocity of its centre as $\overrightarrow{V_O}=4\hat{i}\text{m/s}$. Find the velocity of point $B$ in $\text{m/s}$ on the wheel.

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