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 angular velocity of a wheel is \$70\;rad/s\$. If the radius of the wheel is \$0.5\;m\$, then linear velocity of the wheel is

Given : \$w = 70\$ \$rad/s\$ \$r = 0.5\$ m\$\therefore\$ Its linear velocity \$v = rw = 0.5 \times 70 =35\$ \$m/s\$

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?

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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

A particle is moving in a circle of radius $r$ centred at $O$ with constant speed $v$. What is the change in velocity in moving from $A$ to $B (\angle AOB = 40^{\circ})?$

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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 angular velocity of a wheel is $70\;rad/s$. If the radius of the wheel is $0.5\;m$, then linear velocity of the wheel is

Accepted Answer

Given :     $w  = 70$  $rad/s$  $r  = 0.5$ m$\therefore$ Its linear velocity $v  = rw  = 0.5 \times 70   =35$  $m/s$

Angular Velocity in terms of linear velocity

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