Relation between Linear and Angular Speed

Q. The escape velocity from the Earth's surface is $v.$ The escape velocity from the surface of another planet having a radius, four times that of Earth and same mass density is

1. $4v$
2. $v$
3. $2v$
4. $3v$

Q. The ratio of angular speeds of minute hand and hour hand of a watch is

1. $1:12$
2. $6:1$
3. $12:1$
4. $1:6$

Q.

The length of second's hand in a watch is 1 cm. The change in velocity of its tip in 15 seconds is

1. $Zero$

2. $\frac{\pi }{30\sqrt{2}}cm/sec$

3. $\frac{\pi }{30}cm/sec$

4. $\frac{\pi \sqrt{2}}{30}cm/sec$

Q. A wheel having a radius of $10\text{cm}$ is coupled by a belt to another wheel of radius $30\text{cm}$. If the 1st wheel increases its angular speed from rest at a uniform rate of $1.57{\text{rad/s}}^2$, the time for the second wheel to reach a rotational speed of $100\text{rev/min}$ is (Assume that the belt does not slip)

1. $1.5\text{s}$
2. $20\text{s}$
3. $15\text{s}$
4. $0\text{s}$

Q.

A new unit of length is chosen such that the speed of light in a vacuum is unity. What is the distance between the sun and the earth in terms of the new unit if light takes 8 minutes and 20 seconds to cover this distance?

Q.

A solid sphere is set into motion on a rough horizontal surface with a linear speed v in the forward direction and an angular speed v/R in the anticlockwise direction as shown in figure (10-E16). Find the linear speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts.

Q.

A small object is embedded in a glass sphere $(\mu =1.5)$ of radius 5.0 cm at a distance 1.5 cm left to the centre. Locate the image of the object as seen by an observer standing

(a) To the left of the sphere and

(b) To the right of the sphere.

Q. What is meant by polar and axial vectors?explain.

Q. The linear velocity of a rotating body is given by $\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}$, where $\overrightarrow{\omega }$ is the angular velocity and $\overrightarrow{r}$ is the radius vector. The angular velocity of a body is $\overrightarrow{\omega }=\hat{i}-2\hat{j}+2\hat{k}$ and the radius vector $\overrightarrow{r}=4\hat{j}-3\hat{k}$, then $\left|\overrightarrow{v}\right|$ is

1. $\sqrt{29}$ units
2. $\sqrt{31}$ units
3. $\sqrt{41}$ units
4. $\sqrt{37}$ units

Q. The escape velocity of a body from earth is about $11.2\text{km/s}$. Assuming the mass and radius of the earth to be about $81$ and $4$ times the mass and radius of the moon. The escape velocity in $\text{km/s}$ from the surface of the moon will be

1. $0.54$
2. $2.48$
3. $11$
4. $49.5$

Q.

A solid sphere rolling on a rough horizontal surface with a linear speed v collides elastically with a fixed, smooth, vertical wall. Find the speed of the sphere after it has started pure rolling in the backward direction.

Q. A body is in pure rotation. The linear speed $v$ of the particle on the body, the distance $r$ of the particle from the axis and the angular velocity $\omega$ of the body are related as $\omega =\frac{v}{r}$. Thus

1. $\omega \propto \frac{1}{r}$
2. $\omega \propto r$
3. $v$
4. $\omega$ is independent of $r$

Q. A racing car is travelling along a striaght track at a constant velocity of $40\text{m/s}$. A fixed TV camera is recording the event as shown in figure. In order to keep the car in view, the angular velocity of camera in the position shown should be

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

Q. A particle is moving along a circular path. The angular velocity, linear velocity, angular acceleration, and centripetal acceleration of the particle at any instant respectively are $\overrightarrow{\omega },\overrightarrow{v},\overrightarrow{\alpha }$and $\overrightarrow{a_c}$. Which of the following relations is not correct?

1. $\overrightarrow{\omega }\perp \overrightarrow{v}$
2. $\overrightarrow{\omega }\perp \overrightarrow{\alpha }$
3. $\overrightarrow{\omega }\perp \overrightarrow{a_c}$
4. $\overrightarrow{v}\perp \overrightarrow{a_c}$

Q. Two particles $A$ and $B$ are moving on two concentric circles of radii $R_1$ and $R_2$ respectively with equal angular speed $\omega \text{rad/s}$. At $t=0$, their positions and linear velocities are shown in the figure. Then relative velocity $(\text{m/s}$) of $A$ with respect to $B$ (i.e $\overrightarrow{v_{AB}})$ at $t=\frac{\pi }{2\omega }\text{s}$ is given by:

1. $\omega (R_1+R_2)\hat{i}$
2. $\omega (R_1-R_2)\hat{i}$
3. $\omega (R_2-R_1)\hat{i}$
4. $-\omega (R_1+R_2)\hat{i}$

Q.

The maximum tension in the string of an oscillating pendulum is double of the minimum tension. Find the angular amplitude.

Q. The force required to keep a body in uniform circular motion is

1. resistance.
2. centrifugal force.
3. None of the above
4. centripetal force.

Q.

If the body is moving in a circle of radius r with a constant speed v, its angular velocity is

1. $\frac{v^2}{r}$

2. $vr$

3. $\frac{v}{r}$

4. $\frac{r}{v}$

Q. The planet Mars has two moons, if one of them has a period $7\text{hours},30\text{minutes}$ and an obial radius of $9.0\times {10}^3\text{km}$. Find the mass of Mars.
[Given, $\frac{4{\pi }^2}{G}=6\times {10}^{11}{\text{N}}^{-1}{\text{m}}^{-2}{\text{kg}}^2$]

1. $5.96\times {10}^{19}\text{kg}$
   
2. $3.25\times {10}^{21}\text{kg}$
3. $7.02\times {10}^{25}\text{kg}$
4. $6.00\times {10}^{23}\text{kg}$

Q.

A wheel completes 2000 revolutions to cover the 9.5 km. distance. then the diameter of the wheel is

1. 1.5 m

2. 1.5 cm

3. 7.5 cm

4. 7.5 m

Q. A particle undergoes uniform circular motion. The velocity and angular velocity of the particle at an instant of time are $\overrightarrow{v}=3\hat{i}+4\hat{j}\text{m/s}$ and $\overrightarrow{\omega }=x\hat{i}+6\hat{j}\text{rad/s}$. The value of $x$ in $\text{rad/s}$ is

1. $8$
2. $6$
3. $-8$
4. $-6$

Q. A particle describes a horizontal circle of radius $r$ on the smooth surface of an inverted cone as shown in the figure. The height of plane of circle above vertex is $h$. The speed of the particle should be

1. $\sqrt{rg}$
2. $\sqrt{2rg}$
3. $\sqrt{gh}$
4. $\sqrt{2gh}$

Q.

What is the value of tangential acceleration in uniform circular motion?

Q. One end of the string is pulled with constant velocity $v$ as shown in figure, then velocity of block will be :

1. $v\sin \theta$
2. $v\sec \theta$
3. $v\tan \theta$
4. $\frac{v}{\sin \theta }$

Q. A particle starts from the point $(0,8)m$ and moves with uniform velocity of $3\hat{i}$$m/sec$. After $5sec$, the angular velocity of the particle about the origin will be

1. $\frac{3}{8}rad/\sec$
2. $\frac{8}{17}rad/\sec$
3. $\frac{8}{289}rad/\sec$
4. $\frac{24}{289}rad/\sec$

Q.

A body is whirled in a horizontal circle of radius 20 cm. It has angular velocity of 10 rad/s. What is its linear velocity at any point on circular path

1. 20 m/s

2. $\sqrt{2}$ m/s

3. 2 m/s

4. 10 m/s

Q. A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that:

1. its velocity is constant
2. its acceleration is constant
3. its kinetic energy is constant
4. it moves in a circular path

Q. Consider a uniform rod of mass $M=4m$ and length $L$ pivoted about its centre. A mass $m$ moving with a velocity $V$ making an angle $\theta =\frac{\pi }{4}$ to the rod's long axis collides with one end of the rod, and sticks to it. The angular speed of the rod-mass system just after the collision is:

1. $\frac{3}{7\sqrt{2}}\frac{V}{L}$
2. $\frac{3}{7}\frac{V}{L}$
3. $\frac{3\sqrt{2}}{7}\frac{V}{L}$
4. $\frac{4}{7}\frac{V}{L}$

Q.

A particle undergoes uniform circular motion. about which point on the plane of the circle, will the angular momentum of the particle remain conserved?

1. inside the circle

2. outside the circle

3. center of the circle

4. on the circumference of the circle.

Q. The second's hand of a watch has length 6 cm. Speed of end point and magnitude of difference of velocities at two perpendicular positions will be

1. 6.28 and 0 mm/s
2. 8.88 and 4.44 mm/s
3. 6.28 and 8.88 mm/s
4. 8.88 and 6.28 mm/s