Rotation + Translation

Q. 13. For a particle showing motion under the force F = -5(x-2)2 motion of particle is

Q. a solid sphere of mass m and radius r released from the top of an inclined plane such that it rolls down without slipping if the coefficient of friction is new and angle of inclination is theta with horizontal then force of friction acting on the body is

Q. if 5 small spherical droplet coalesce to form a bigger drop then the temperature of bigger drop in comparison to small drop will be

Q. A wheel of radius $r=1\text{m}$ rolls without slipping with an angular velocity of $\omega =\frac{\pi }{4}\text{rad/s}$ about its centre and velocity of its centre of mass is $v_{COM}=2\text{m/s}$. If the point $P(0,0)$ is in contact with the ground initially, find its position vector after time $t=1\text{s}$.

1. $(1-\frac{1}{\sqrt{2}})\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$
2. $(1-\frac{1}{2})\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$
3. $\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$
4. $(2-\frac{1}{\sqrt{2}})\hat{i}+(1-\frac{1}{\sqrt{2}})\hat{j}$

Q. A hollow sphere rolls without slipping on the horizontal surface such that its translational velocity is $v$. Find that the maximum height attained by it on an inclined surface as shown in the figure.

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

Q. A solid cylinder of mass $m$ and radius $r$ starts rolling down on inclined plane of inclination $\theta$. Friction is sufficient enough to prevent slipping. Find the speed of centre of mass of solid cylinder, when it's centre of mass has fallen through a height of $h=9\text{m}$. (Take $g=10{\text{m/s}}^2$)

1. $\sqrt{120}\text{m/s}$
2. $\sqrt{125}\text{m/s}$
3. $\sqrt{180}\text{m/s}$
4. $\sqrt{90}\text{m/s}$

Q. 147 a small solid. marble of mass M and radius r rolls down an inclined plane with a loop tack of radius R at the end without slipping . find the height h above the base from where it has to start rolling down the incline such that sphere just completes the vertical loop

Q. Two blocks A and B of equal masses are sliding down along straight parallel lines on an inclined plane of 45^° . Theircoefficients of kinetic friction are u = 0.2 and H0.3 respectively. At t = 0, both the blocks are at rest and block A is2 meter behind block B. The time and distance from the initial position where the front faces of the blocks come in2.line on the inclined plane as shown in figure. (Use g\lbrack JEE 2004 (Scr.) 3/84110 ms2.)2mBFixed45(A) 2s, 8/2m(B) 2 s, 7m(C) 2 s, 7/2m(D) 2s, 7//2 mA disc is kent on a

Q. a ball reaches a racket at 60m/s along +x axis direction, and leaves the racket in the opposite direction with the same speed .assuming that the mass of the ball as 50gm and exerted by the racket on the ball is

Q. A particle is placed at rest inside a hollow hemisphere of radius $R$. The coefficient of friction between particle and hemisphere is $\mu =\frac{1}{\sqrt{3}}$ . The maximum height up to which particle can remain stationary.

1. $[1+\frac{\sqrt{3}}{2}]R$
2. $[1-\frac{\sqrt{3}}{2}]R$
3. $\frac{R}{2}$
4. $\frac{\sqrt{3}R}{2}$

Q. what is the velocity of a ball thrown upward and why?

Q. A uniform rod of mass $m$ and length $l$ is released from rest in the vertical position on a rough (surface is sufficiently rough to prevent sliding) square corner A, shown in figure, then choose correct options: (given that normal acts along the length of the rod)

1. If the rod just begins to slip when $\theta ={37}^{\circ }$ with vertical, then the coefficient of static friction $\mu$ between the rod and the corner is 0.3.
2. If the end of the rod is notched so that it cannot slip, then the angle $\theta$ at which contact between the rod and the corner ceases is ${53}^{\circ }$
3. If the rod just begins to slip when $\theta ={37}^{\circ }$ , then the coefficient of static friction $\mu$ between the rod and the corner is $\frac{3}{4}$
4. If the end of the rod is notched so that it cannot slip, then the angle $\theta$ at which contact between the rod and the corner ceases is ${37}^{\circ }$.

Q. A hollow sphere of mass $M$ and radius $R$ is initially at rest on a horizontal rough surface. It moves under the action of a constant horizontal force $F$ as shown in figure.


The linear acceleration $\left(a\right)$ of the sphere is :

1. $a=\frac{10F}{7M}$
2. $a=\frac{7F}{5M}$
3. $a=\frac{6F}{5M}$
4. $a=\frac{F}{M}$

Q. A circular road of radius 50 m has the angle of banking equal to 30°. At what speed should a vehicle go on this road so that the friction is not used?

Q. What is relative motion

Q. 47.a person with heavy weight in his hands close to the chest is standing on the rotating table when he stretches his hands his speed of rotation decreases explain why

Q. A man moving with a velocity of $5\text{m/s}$ on a horizontal road observes that raindrops fall at an angle of ${45}^{\circ }$ with the vertical. When he moves with a velocity of $16\text{m/s}$ along an inclined plane, which is inclined at ${30}^{\circ }$ with the horizontal, he observes raindrops falling vertically downward as shown in the figure. Find the actual velocity ${\overrightarrow{V}}_r$ of the raindrops.

1. $\overrightarrow{V_r}=\sqrt{3}\hat{i}-(\sqrt{3}-5)\hat{j}$
2. $\overrightarrow{V_r}=4\sqrt{3}\hat{i}-(4\sqrt{3}-5)\hat{j}$
3. $\overrightarrow{V_r}=8\sqrt{3}\hat{i}-(8\sqrt{3}-5)\hat{j}$
4. $\overrightarrow{V_r}=2\sqrt{3}\hat{i}-(2\sqrt{3}-5)\hat{j}$

Q. A point $P$ (on a wheel) is the contact point of the wheel and the ground. The wheel rolls on the ground without slipping. The value of displacement of the point $P$, when the wheel completes half a rotation is (if radius of wheel is $1\text{m}$)

1. $2\text{m}$
2. $\sqrt{{\pi }^2+4}\text{m}$
3. $\pi \text{m}$
4. $\sqrt{{\pi }^2+2}\text{m}$

Q. A solid cylinder of radius R is kept at rest on a smooth horizontal surface. A horizontal force is applied to move the cylinder under pure rolling. Then height of force from horizontal surface is does smooth and rough surface change the height for pure rolling to happen??

Q.

Two point A and B Lie on two extremes of a diameter AB as shown in figure. The sphere rotates with constant angular velocity $\omega$ about an axis as shown. What is the angular velocity of Q with respect to P?

1. 

2. 

3. 2

4. 

Q. A juggler throws ball into air. He throws one whenever the previous one is at its highest point. How high do the ball rise if he throws n balls in each second (Neglecting air friction).

Q. A solid sphere is rolling without slipping on a frictionless surface with a translational velocity $v\text{m/s}$ as shown in the figure. If it has to climb the inclined surface then $v$ in $\text{m/s}$ should be:

1. $\ge \sqrt{\frac{10gh}{7}}$
   
2. $\le \sqrt{2gh}$
   
3. $\ge \sqrt{\frac{5gh}{7}}$
   
4. $\le \frac{17}{7}\sqrt{gh}$
   
   

Q. Derive the position of centre of mass of an uniform hollow cone.

Q. A solid cylinder is released from rest from the top of an inclined plane of inclination $\theta$ and length ${}^{\prime }{l}^{\prime }$. If the cylinder rolls without slipping, then find it's speed when it reaches the bottom of inclined plane.

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

Q.

Two cars P and Q move with velocities ${\overrightarrow{v}}_P$ and ${\overrightarrow{v}}_Q{\overrightarrow{v}}_Q$ towards north and east respectively.

Velocity of car P w.r.t. Q (i.e., velocity of car P for an observer on cars Q) is ${\overrightarrow{v}}_{AB}$ will be in which direction?

1. South - east

2. North - west

3. North - east

4. South - west

Q. A solid sphere of mass $500\text{g}$ and radius $10\text{cm}$ rolls without slipping with a velocity of $20\text{cm/s}$. The total $K.E$. of the sphere is nearly

1. $0.001\text{J}$
2. $0.014\text{J}$
3. $0.14\text{J}$
4. $1.4\text{J}$

Q. A disc of mass $M$ and radius $R$ is rolling without slipping with angular speed $\omega$ on a horizontal plane as shown in figure. The magnitude of angular momentum of the disc about the $x-$axis is:

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

Q. A cord is wound round the circumference of wheel of radius 'r'. The axis of the wheel is horizontal and moment of inertia about it is 'I'. A weight 'mg' is attached to the end of the cord and falls from rest. After falling through a distance h, the angular velocity of the wheel will be

1. $\sqrt{\frac{2gh}{I+mr}}$
2. $\sqrt{\frac{2mgh}{I+m{r}^2}}$
3. $\sqrt{\frac{2mgh}{I+2m{r}^2}}$
4. $\sqrt{2gh}$

Q.

Two point A and B Lie on two extremes of a diameter AB as shown in figure. The sphere rotates with constant angular velocity $\omega$ about an axis as shown. What is the angular velocity of Q with respect to P?

1. $\frac{\omega }{2}$

2. $\frac{\omega }{4}$

3. 2 $\omega$

4. $\omega$

Q. If a body does not respond to any amount of external force applied to it in any way, then only it is rigid.

1. True
2. False