Instantaneous Axis of Rotation

Q.

A small roller of diameter $20cm$ has an axle of diameter $10cm$ (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved$50cm$, the position of the scale will look like (figures are schematic and not drawn to scale)

1. 

2. 

3. 

4. 

Q. A uniform rod $AB$ of mass $M$ and length $L$ is hinged at end $A$. It is released from rest at a horizontal position. The linear acceleration of the centre of mass immediately after release is

1. g
2. $\frac{g}{2}$
3. $\frac{g}{4}$
4. $\frac{3g}{4}$

Q. The motor of an engine is rotating its axis with an angular velocity of $100\text{rev per min}$. It comes to rest in $15\text{s}$ after being switched off, assuming constant angular deceleration. What is the number of revolutions made by it before coming to rest ?

1. $12.5$
2. $40$
3. $32.6$
4. $15.6$

Q.

A particle falls towards earth from infinity. Its velocity on reaching the earth would be

1. Infinity

2. $\surd 2gR$

3. $2\surd gR$

4. Zero

Q. Two small rings $O$ and $O^{\prime }$ are put on two vertical stationary rods $AB$ and $A^{\prime }{B}^{\prime }$, respectively. One end of an inextensible thread is tied at a point $A^{\prime }$. The thread passes through ring $O^{\prime }$ and its other end is tied to ring $O$. Assuming that ring $O^{\prime }$ moves downwards at a constant velocity $v_1$, then velocity $v_2$ of the ring
$O$, when $\angle AO{O}^{\prime }$= $\alpha$, is

1. None of these
2. $v_1\left[\frac{2{\sin }^2\alpha /2}{\cos \alpha }\right]$
3. $v_1\left[\frac{2{\cos }^2\alpha /2}{\sin \alpha }\right]$
4. $v_1\left[\frac{3{\cos }^2\alpha /2}{\sin \alpha }\right]$

Q. A car moves with constant tangential acceleration a(t)=0.80m/S2 along a horizontal surface circumscribing a circle of radius R=40m. The coefficient of sliding friction between the wheels of the car and the surface is Meu=0.20. What distance will the car ride without sliding of its initial speed is 0.

Q. A ladder of length $L$ is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment, the speed of the end in contact with the horizontal floor is $v$ and the ladder makes an angle $\theta ={30}^{\circ }$ with horizontal. Then, the speed of the ladder's centre of mass must be-

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

Q. In the shown figure, the strings are massless. If the vertical rod is rotated the tension in upper string is $35\text{N}$ then the speed of the ball is

1. $5.4\text{m/s}$
2. $4.4\text{m/s}$
3. $3.4\text{m/s}$
4. $6.4\text{m/s}$

Q. A solid sphere of mass 5 kg and radius 10 m is rolling purely on a horizontal surface as shown in the figure. The separation of P from A (point of contact) is 7 m. If the sphere has an angular velocity about an axis through its centre of mass as 2 $\frac{rad}{sec},$ what is the speed of the point P at this instant ?

1. 20 m/s
2. 32 m/s
3. 14 m/s
4. 18 m/s

Q. A plank is kept over a cylinder as shown in figure and there is no slipping at any contact. $15\text{N}$ is the horizontal force applied on the plank. Then, identify the correct statement(s):

1. Acceleration of plank is $\frac{60}{7}{\text{m/s}}^2$
2. Acceleration of centre of mass of the cylinder is $\frac{30}{7}{\text{m/s}}^2$
3. Friction force between plank and cylinder is $\frac{45}{7}\text{N}$
4. Friction force between cylinder and ground is $\frac{15}{7}\text{N}$

Q. A hollow cylinder is rolling without slipping on a horizontal surface as shown in figure. Then, the angular acceleration of the cylinder will be -

1. $10{\text{rad/s}}^2$
2. $16{\text{rad/s}}^2$
3. $20{\text{rad/s}}^2$
4. $12{\text{rad/s}}^2$

Q. A solid sphere of mass $0.50\text{kg}$ is kept at rest on horizontal surface. The coefficient of static friction between the solid sphere and the surface in contact is $\frac{2}{7}$. The maximum force that can be applied at the highest point of sphere in the horizontal direction so that the sphere dosen't slip on the surface is given by $\frac{x}{3}\text{N}$, then value of $x$ is
(Assume $g=10{\text{ms}}^{-2}$)

Q. End $A$ of a rod $AB$ is being pulled on the floor with a constant velocity $v_0$ as shown. Taking the length of the rod as $l$, at an instant when the rod makes an angle ${37}^{\circ }$ with the horizontal, calculate


The velocity of end $B$

1. $\frac{3}{5}{v}_0$
2. $\frac{4}{3}{v}_0$
3. $\frac{5}{3}{v}_0$
4. $\frac{5}{4}{v}_0$

Q. A disc of radius $0.2\text{m}$ is rolling with slipping on a flat horizontal surface as shown in the figure. The instantaneous centre of rotation is (the lowest contact point is O and centre of disc is C)

1. zero
2. $0.1\text{m}$ above O on line OC
3. $0.2\text{m}$ below O on line OC
4. $0.2\text{m}$ above O on line OC

Q.

A diver having a moment of inertia of 6.0 ${\text{kg-m}}^2$ about an axis through its centre of mass rotates at an angular speed of 2 rad/s about this axis. If he folds his hands and feet to decrease the moment of inertia to 5.0 ${\text{kg-m}}^2$, what will be the new angular speed ?

Q. A plank is moving with a velocity of $4\text{m/s}$. A disc of radius $1\text{m}$ rolls without slipping on it with an angular velocity of $3\text{rad/s}$ as shown in figure. Find out the velocity of the centre of the disc.

1. $3\text{m/s}$
2. $4\text{m/s}$
3. $7\text{m/s}$
4. $11\text{m/s}$

Q. A uniform rod of length $L$ (in between the supports) and mass m is placed on two hinges A and B. The rod breaks suddenly at length $\frac{L}{10}$ from the support B. Find the reaction at hinge A immediately after the rod breaks :

1. $\frac{9}{40}\text{mg}$
2. $\frac{19}{40}\text{mg}$
3. $\frac{\text{mg}}{2}$
4. $\frac{9}{20}\text{mg}$

Q.

A disc rotating about its axis with angular speed ${\omega }_0$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. What are the linear velocities of the points A, B and C on the disc shown in Fig. 7.41? Will the disc roll in the direction indicated?

Q. The stick applies a force of $2\text{N}$ on the ring$(R=0.5\text{m})$ and the ring rolls without slipping on the ground as shown in figure. The coefficient of friction between the stick and the ring is

1. $0.4$
2. $0.3$
3. $0.2$
4. $0.5$

Q. A plank is kept over a cylinder as shown in figure and there is no slipping at any contact. $15\text{N}$ is the horizontal force applied on the plank. Then, identify the correct statement(s):

1. Acceleration of plank is $\frac{60}{7}{\text{m/s}}^2$
2. Acceleration of centre of mass of the cylinder is $\frac{30}{7}{\text{m/s}}^2$
3. Friction force between plank and cylinder is $\frac{45}{7}\text{N}$
4. Friction force between cylinder and ground is $\frac{15}{7}\text{N}$

Q. A particle is projected with a velocity u at an angle (theta) with the horizontal. Find the radius of curvature of the parabola traced out by the particle at the point where its velocity makes an angle (theta/2) with the horizontal.

Q. A uniform sphere of mass m is positioned between two horizontal planks A and B of mass 2 m and m respectively as shown. Ground surface is smooth and friction between planks and sphere is sufficient so that no slipping takes place. Horizontal forces $F_1$ and $F_2$ are applied on plank so that acceleration of sphere is zero. Plank B is moving towards right with acceleration $a$. Which of the following is/are correct?

1. $\text{Both planks have same acceleration}$
2. $\text{Net force on sphere is non-zero}$
3. $F_1+F_2=\frac{17}{5}ma$
4. $\text{friction}=5ma$

Q. Find the instantaneous axis of rotation of a rod of length $l$ from the end $A$ when it moves with a velocity $\overrightarrow{v_A}=v\hat{i}$ and the rod rotates with an angular velocity $\overrightarrow{\omega }=-\frac{v}{2l}\hat{k}$, shown in the figure.

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

Q. A uniform solid cylinder of mass $m$ and radius $R$ is placed on a rough horizontal surface A horizontal constant force $F$ is applied at the top point $P$ of the cylinder so that it starts pure rolling. The frictional force on the cylinder is :

1. zero.
2. $\frac{F}{3}$, towards left
3. $\frac{2F}{3}$, towards right
4. $\frac{2F}{3}$, towards left

Q. Two billiards ball are rolling on a flat table. one has the velocity components Vx = 1m/s, Vy =3 m/s and the other one has components Vx = 2m/s and Vy= 2 m/s . If both the balls start rolling from the same point, what is the angle between their paths?

Q. A disc which is at rest started to spin about a stationary axis. The angular acceleration of the disc is given by $\alpha =2\theta rad/s$, where $\theta$ is the angle of rotation of disc from its initial position. Which of the following graphs correctly represents the variation of angular velocity with respect to '$\theta$' ?

1. 
2. 
3. 
4. 

Q. End $A$ of a rod $AB$ is being pulled on the floor with a constant velocity $v_0$ as shown. Taking the length of the rod as $l$, at an instant when the rod makes an angle ${37}^{\circ }$ with the horizontal, calculate

The angular velocity of the rod

1. $\frac{5v_0}{3l}$
2. $\frac{3v_0}{5l}$
3. $\frac{5v_0}{4l}$
4. $\frac{4v_0}{5l}$

Q. A small uniform rod AB of mass M and length l rotates freely with angular velocity ${\omega }_0$ in a horizontal plane about a vertical axis passing through one of its end A. A small sleeve of mass M starts sliding along the rod from A. Angular velocity of rod when sleeve reaches the other end, is

1. $\frac{{\omega }_0}{4}$
2. $\frac{2{\omega }_0}{3}$
3. $\frac{13{\omega }_0}{15}$
4. $\frac{{\omega }_0}{3}$

Q. End $A$ of a rod $AB$ is being pulled on the floor with a constant velocity $v_0$ as shown. Taking the length of the rod as $l$, at an instant when the rod makes an angle ${37}^{\circ }$ with the horizontal, calculate



The velocity of the CM of the rod

1. $\frac{5}{7}{v}_0$ at $ta{n}^{-1}\frac{4}{3}$ below horizontal
2. $\frac{5}{7}{v}_0$ at $ta{n}^{-1}\frac{3}{4}$ below horizontal
3. $\frac{5}{6}{v}_0$ at $ta{n}^{-1}\frac{3}{4}$ below horizontal
4. $\frac{4}{5}{v}_0$ at $ta{n}^{-1}\frac{3}{4}$ below horizontal

Q. A rod of length $2m$ is at a temperature of ${20}^{o}C$. Find the free expansion of the rod, if the temperature is increased to ${50}^{o}C$, then find stress produced when the rod isFully prevented to expand

1. $5\times {10}^{7}N/m^2$
2. $18\times {10}^{8}N/m^2$
3. $18\times {10}^{7}N/m^2$
4. $9\times {10}^{8}N/m^2$