Relation between Linear and Angular Speed

Q.

A scooter is going round a circular road of radius 100 m at a speed of 10 m/s. The angular speed of the scooter will be

1. 0.01 rad/s

2. 0.1 rad/s

3. 1 rad/s

4. 10 rad/s

Q. A circular disc is rotating with constant $\omega =2$ rad/sec, about an axis passing through its centre. Two objects are kept at a distance of ${}^{\prime }{r}_1=2c{m}^{\prime }$ and ${}^{\prime }{r}_2=1c{m}^{\prime }$ from the centre. The ratio of their linear speeds is.

1. 2:1
2. 4:1
3. 1:1
4. 2:4

Q. A body is projected with initial velocity of $(8\hat{i}+6\hat{j})\text{m/s}$ from origin, find angular velocity of body wrt to an axis passing through origin and perpendicular to the x -y plane when it lands. (Here $\hat{i},\hat{j}$ are unit vectors corresponding to X, Y axis respectively)
Take $g=10{\text{m/s}}^2$

1. $\frac{5}{8}rad/s$
2. $\frac{8}{5}rad/s$
3. $2.4rad/s$
4. $4.5rad/s$

Q. This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : Linear momentum of a body changes even when it is moving uniformly in a circle.
A : एक पिण्ड का रेखीय संवेग तब भी परिवर्तित होता है, जब यह एक वृत्त में एकसमान रूप से गतिशील हो।

R : In uniform circular motion, velocity remains constant.
R : एकसमान वृत्तीय गति में, वेग नियत रहता है।

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   (A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. A particle moves in a circular path of radius $40\text{cm}$ as shown. Find angular velocity of particle wrt an axis passing through centre of the circle and perpendicular to plane of the circle.

1. $15\text{rad/s}$
2. $12\text{rad/s}$
3. $16\text{rad/s}$
4. $14\text{rad/s}$

Q. Two particles $A$ and $B$ are moving with equal angular speed $\omega$ as shown in figure. At $t=0$, their positions are shown. Then, relative velocity $\overrightarrow{v_A}-\overrightarrow{v_B}$ at $t=\frac{\pi }{2\omega }$ is

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

Q. A bucket is being lowered down into a well through a rope passing over a fixed pulley of radius $10\text{cm}$. Assume that the rope does not slip on the pulley. Find the angular velocity of the pulley at an instant when the bucket is going down at a speed of $20\text{cm/s}$ and has an acceleration of $4.0{\text{m/s}}^2$

1. $2\text{rad/s}$
2. $0.2\text{rad/s}$
3. $0.02\text{rad/s}$
4. $20\text{rad/s}$

Q. In the figure shown below, $\overrightarrow{\omega }$ is angular velocity and $\overrightarrow{r}$ is position vector. What is the direction and magnitude of linear velocity.

1. $\hat{k},20\text{m/s}$
2. $-\hat{k},20\text{m/s}$
3. $\hat{k},\frac{20}{\sqrt{2}}\text{m/s}$
4. $-\hat{k},\frac{20}{\sqrt{2}}\text{m/s}$

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. 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. $\sqrt{37}$ units
4. $\sqrt{41}$ units

Q.

A scooter is going round a circular road of radius 100 m at a speed of 10 m/s. The angular speed of the scooter will be

1. 0.01 rad/s

2. 0.1 rad/s

3. 1 rad/s

4. 10 rad/s

Q. A car moves on a circular road, describing equal angles about the centre in equal intervals of time. Which of the following statements about the velocity of the car are not true?

1. Velocity is constant.
2. Magnitude of velocity is constant but the direction changes.
3. Both magnitude and direction of velocity change.
4. Velocity is directed towards the centre of circle.

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.

What is the value of linear velocity, if $\overrightarrow{\omega }=3\hat{i}-4\hat{j}+\hat{k}$ and $\overrightarrow{r}=5\hat{i}-6\hat{j}+6\hat{k}$

1. $6\hat{i}+2\hat{j}-3\hat{k}$

2. $-18\hat{i}-13\hat{j}+2\hat{k}$

3. $4\hat{i}-13\hat{j}+6\hat{k}$

4. $6\hat{i}-2\hat{j}+8\hat{k}$

Q. Two points $P$ and $Q$ are diametrically opposite on a disc of radius $R$. A particle at point $P$ has linear velocity $v$ as shown in figure. Find the angular speed of particle wrt an axis passing through $Q$ and perpendicular to plane of the disc.

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

Q. Assertion: Vector product of two vectors is an axial vector.
Reason: For a body under rotation, if $\overrightarrow{v}=$ instantaneous velocity, $\overrightarrow{r}=$ radius vector and $\overrightarrow{\omega }=$ angular velocity, then $\overrightarrow{\omega }=\overrightarrow{v}\times \overrightarrow{r}$.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true and reason is not the correct explanation of assertion.
3. If assertion is true but reason is false
4. If both assertion and reason are false

Q. A particle A moves along a circle of radius $R=10cm$ so that its radius vector $\overrightarrow{r}$ relative to O rotates with constant angular velocity $\omega =0.2rad/s$. Find the magnitude of the velocity of the particle and its total acceleration.

1. $4cm/s,1.6cm/s^2$
2. $4cm/s,2.6cm/s^2$
3. $12cm/s,1.6cm/s^2$
4. $12cm/s,1.8cm/s^2$

Q. The ball is rotated on a horizontal circular path about the vertical axis as shown in figure. The maximum tension that the string can bear is $500\text{N}$. Then choose the correct option(s):
[Take $g=10{\text{m/s}}^2$]

1. Maximum possible value of angular velocity of ball is $5\text{rad/s}$.
2. Maximum possible value of angular velocity of ball is $8\text{rad/s}$.
3. Maximum possible value of linear velocity of ball is $5\sqrt{6}\text{m/s}$.
4. Maximum possible value of linear velocity of ball is $4\sqrt{6}\text{m/s}$.

Q.

A circular disc is rotating with constant $\omega =2$ rad/sec, about an axis passing through its centre. An object is kept at a distance of ${}^{\prime }{r}_1=2c{m}^{\prime }$ from the centre and another object is kept at a distance of ${}^{\prime }{r}_2=1c{m}^{\prime }$. Find the ratio of their linear speeds.

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

Q. Angular velocity of a body moving on a circular track is given by $\overrightarrow{\omega }=\hat{i}+\hat{j}+\hat{k}$ and the radius vector $\overrightarrow{r}=2\hat{i}-2\hat{j}-2\hat{k}$ at the instant of time $‘t^{\prime }$. Then the magnitude of linear velocity of the body at that instant is.

1. $2.01\text{m/s}$
2. $4.23\text{m/s}$
3. $6.21\text{m/s}$
4. $5.65\text{m/s}$

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

Q. In the figure shown below, $\overrightarrow{\omega }$ is angular velocity and $\overrightarrow{r}$ is position vector. What is the direction and magnitude of linear velocity.

1. $\hat{k},20\text{m/s}$
2. $-\hat{k},20\text{m/s}$
3. $\hat{k},\frac{20}{\sqrt{2}}\text{m/s}$
4. $-\hat{k},\frac{20}{\sqrt{2}}\text{m/s}$

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. $\frac{r}{v}$