Q: Angular velocity of minute hand of a clock is

Time taken by minute hand to rotate by \$2\pi\$ radians \$ t =1\$ \$hr = 3600\$ s\$\therefore\$ Its angular velocity \$w = \dfrac{\theta}{t} = \dfrac{2\pi}{3600} =\dfrac{\pi}{1800}\$ \$rad/s\$

Q: Two particles having mass \$M\$ and \$m\$ are moving in a circular path having radius \$R\$ and \$r\$. If their time period are same, then the ratio of their angular velocity will be :

Time period of revolution \$T = \dfrac{2\pi}{w}\$As according to question, \$T_1 = T_2\$\$\therefore\$ \$\dfrac{2\pi}{w_1} = \dfrac{2\pi}{w_2}\$ \$\implies w_1 = w_2\$

Q: A body is whirled in a horizontal circle of radius \$20\$cm. It has angular velocity if \$10\$ rad/s. What is the linear velocity at any point on the circular path?

Given,Radius of circle, \$r=20\,cm=0.2\,m\$Angular velocity, \$\omega=10\,rad/s\$Linear velocity, \$v=r\times \omega=0.2\times 10=2\,m/s\$

Q: A car has wheels of radius \$0.30\ m\$ and is travelling at \$36\ m/s\$. Calculate :(a) the angular speed of the wheel.(b) If the wheels describe \$40\$ revolutions before coming to rest with a uniform acceleration.(i) find it angular acceleration and(ii) the distance covered.

Radius \$r=0.30m\$, Velocity, \$v=36m/s\$ \$(a)\$ angular speed \$w=\cfrac { v }{ r } \\ \quad =\cfrac { 36 }{ 0.3 } =120\quad rad/s\$ \$(b)\$ final angular speed \$w'=0\$ From rotational kinematics \${ w }^{ '2 }-{ w }^{ 2 }=2\alpha \theta \\ 0-14400=2\alpha \times 40\times 2\pi \\ \Rightarrow \alpha =\cfrac { -14400 }{ 160\pi } =-28.6rad/{ s }^{ 2 }\$ 'Distance covered \$=40\times 2\pi r\\ =40\times 2\pi \times 0.3=75.39m\$

Q: The centre of a wheel rolling on a plane surface moves with a speed \$v_0\$. A particle on the rim of the wheel at the same level as the centre will be moving at speed

As per question,The speed of the centre of wheel \$=v_o\$Particle is on the rim of the wheel,So,The situation can be shown as in the diagramHere \$v_0=R\omega\$At \$P,v=r\omega\$\$=\sqrt{(R^2+R^2)\omega}=\sqrt{2}R\omega=\sqrt{2}v_0\$So the speed of the particle is on the rim \$=\sqrt2R\omega\$

Question 1

Obtain the relation between linear velocity and angular velocity

Accepted Answer

Consider a particle moving with uniform circular motion in anticlock wise direction with centre $O$ and radius $r$ the particle cover an arc of length $\triangle s$ in time $\triangle t$ moving from $A$ to $B$.Hence the angular displacement is$\triangle \theta = \dfrac {\triangle s}{r}$Dividing both side by $\triangle t$$\dfrac {\triangle \theta}{\triangle t} = \dfrac {1}{r} \dfrac {\triangle s}{\triangle t}$In the time interval $\triangle t$ be infinitesimally small $(\triangle t \rightarrow 0)$, then$\displaystyle \lim_{\triangle t \rightarrow \theta} \dfrac {\triangle \theta}{\triangle t} = \dfrac {1}{r} \left [\displaystyle \lim_{\triangle t \rightarrow \theta} \dfrac {\triangle s}{\triangle t}\right ]$But $\displaystyle \lim_{\triangle t \rightarrow \theta} \dfrac {\triangle \theta}{\triangle t} = \omega$ and $\displaystyle \lim_{\triangle t \rightarrow \theta} \dfrac {\triangle s}{\triangle t} = V$$\therefore \omega = \dfrac {V}{r}$$V = r\omega$In vector form, $\vec {V} = \vec {\omega}\times \vec {r}$

Question 2

Angular velocity of minute hand of a clock is

Accepted Answer

Time taken by minute hand to rotate by $2\pi$ radians      $ t =1$ $hr  = 3600$  s$\therefore$   Its angular velocity      $w  = \dfrac{\theta}{t}  = \dfrac{2\pi}{3600}   =\dfrac{\pi}{1800}$    $rad/s$

Question 3

Two particles having mass $M$ and $m$ are moving in a circular path having radius $R$ and $r$. If their time period are same, then the ratio of their angular velocity will be :

Accepted Answer

Time period of revolution        $T  = \dfrac{2\pi}{w}$As according to question,           $T_1  = T_2$$\therefore$     $\dfrac{2\pi}{w_1}  = \dfrac{2\pi}{w_2}$                    $\implies w_1  = w_2$

Question 4

A stone tied to the end of a string of 1 m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolution in 44 seconds, what would be the magnitude and direction of acceleration of the stone?

Accepted Answer

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Question 5

Derive the relation between linear and angular velocity,or Derive $ v=rw $

Accepted Answer

Consider a body rotating about an axis passing through a point $ O $and perpendicular to the plane of the paper. Let $P$ be the position of a particle inside the body. If the body rotates through an angle $ 0 $ in a time interval of $ t $, the particle at $P$ reaches $P ' $ $ \therefore $  Linear displacement $PP' =r \theta $ $ \therefore $Linear velocity, $ v=\dfrac{PP'}{t}=\dfrac{r \theta}{t } $But $ \dfrac{\theta}{t}=\omega $, the angular velocity$ \therefore $Linear velocity , $ v=r \omega $

Question 6

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

Accepted Answer

Given,Radius of circle, $r=20\,cm=0.2\,m$Angular velocity, $\omega=10\,rad/s$Linear velocity, $v=r\times \omega=0.2\times 10=2\,m/s$

Question 7

The angular velocity (in rad/s) of a body rotating at N r.p.m. is

Accepted Answer

Angular velocity is defined as the rate of change of angular displacement with respect to time. It is usually expressed by a Greek letter  (omega). Mathematically, angular velocity is $\omega =d\theta/dt$If a body is rotating at the rate of N r.p.m. (revolutions per minute), then its angular velocity,$\omega=\dfrac{2 \pi N}{60} $rad/s

Question 8

In a rotational motion of a rigid body, all particles move with _____.

Accepted Answer

In rotational motion of the rigid body, all the particles cover the same angular displacement in a particular interval. So, the angular velocities of all the particles will be the same.But the linear velocity is also dependent on the distance of particles from axis of rotation. So, linear velocities will be different for all particles as the distances are different for all particles..Option (C) is correct.

Question 9

A car has wheels of radius $0.30\ m$ and is travelling at $36\ m/s$. Calculate :(a) the angular speed of the wheel.(b) If the wheels describe $40$ revolutions before coming to rest with a uniform acceleration.(i) find it angular acceleration and(ii) the distance covered.

Accepted Answer

Radius $r=0.30m$, Velocity, $v=36m/s$&#10;&#10;$(a)$ angular speed $w=\cfrac { v }{ r } \ \quad =\cfrac { 36 }{ 0.3&#10;} =120\quad rad/s$&#10;&#10;$(b)$ final angular speed $w'=0$&#10;&#10;From rotational kinematics ${ w }^{ '2 }-{ w }^{ 2 }=2\alpha \theta \&#10;0-14400=2\alpha \times 40\times 2\pi \ \Rightarrow \alpha =\cfrac { -14400 }{&#10;160\pi  } =-28.6rad/{ s }^{ 2 }$&#10;&#10;'Distance covered&#10;&#10;$=40\times 2\pi r\ =40\times 2\pi \times 0.3=75.39m$&#10;&#10;

Question 10

The centre of a wheel rolling on a plane surface moves with a speed $v_0$. A particle on the rim of the wheel at the same level as the centre will be moving at speed

Accepted Answer

As per question,The speed of the centre of wheel $=v_o$Particle is on the rim of the wheel,So,The situation can be shown as in the diagramHere $v_0=R\omega$At $P,v=r\omega$$=\sqrt{(R^2+R^2)\omega}=\sqrt{2}R\omega=\sqrt{2}v_0$So the speed of the particle is on the rim $=\sqrt2R\omega$

Revise with Concepts

Angular Velocity and its Relation With Linear Velocity

Quick Summary With Stories

Angular Velocity in terms of Linear Velocity

Instantaneous Angular Velocity

Obtain the relation between linear velocity and angular velocity

Angular velocity of minute hand of a clock is

Two particles having mass $M$ and $m$ are moving in a circular path having radius $R$ and $r$ . If their time period are same, then the ratio of their angular velocity will be :

A stone tied to the end of a string of 1 m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolution in 44 seconds, what would be the magnitude and direction of acceleration of the stone?

Derive the relation between linear and angular velocity,or Derive $v = r w$

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

The angular velocity (in rad/s) of a body rotating at N r.p.m. is

In a rotational motion of a rigid body, all particles move with _____.

A car has wheels of radius $0.30 m$ and is travelling at $36 m / s$ . Calculate :
(a) the angular speed of the wheel.
(b) If the wheels describe $40$ revolutions before coming to rest with a uniform acceleration.
(i) find it angular acceleration and
(ii) the distance covered.

The centre of a wheel rolling on a plane surface moves with a speed $v_{0}$ . A particle on the rim of the wheel at the same level as the centre will be moving at speed

Revise with Concepts

Angular Velocity and its Relation With Linear Velocity

Quick Summary With Stories

Angular Velocity in terms of Linear Velocity

Instantaneous Angular Velocity

Obtain the relation between linear velocity and angular velocity

Angular velocity of minute hand of a clock is

Two particles having mass M and m are moving in a circular path having radius R and r. If their time period are same, then the ratio of their angular velocity will be :

A stone tied to the end of a string of 1 m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolution in 44 seconds, what would be the magnitude and direction of acceleration of the stone?

Derive the relation between linear and angular velocity,or Derive v=rw

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

The angular velocity (in rad/s) of a body rotating at N r.p.m. is

In a rotational motion of a rigid body, all particles move with _____.

A car has wheels of radius 0.30 m and is travelling at 36 m/s. Calculate : (a) the angular speed of the wheel. (b) If the wheels describe 40 revolutions before coming to rest with a uniform acceleration. (i) find it angular acceleration and (ii) the distance covered.

The centre of a wheel rolling on a plane surface moves with a speed v0​. A particle on the rim of the wheel at the same level as the centre will be moving at speed

Two particles having mass $M$ and $m$ are moving in a circular path having radius $R$ and $r$ . If their time period are same, then the ratio of their angular velocity will be :

Derive the relation between linear and angular velocity,or Derive $v = r w$

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

A car has wheels of radius $0.30 m$ and is travelling at $36 m / s$ . Calculate :
(a) the angular speed of the wheel.
(b) If the wheels describe $40$ revolutions before coming to rest with a uniform acceleration.
(i) find it angular acceleration and
(ii) the distance covered.

The centre of a wheel rolling on a plane surface moves with a speed $v_{0}$ . A particle on the rim of the wheel at the same level as the centre will be moving at speed