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>>Angular Velocity and its Relation with Linear Velocity
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Angular Velocity and its Relation With Linear Velocity

Q: Obtain the relation between linear velocity and angular velocity

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}\$

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

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 \$

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

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

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\$

Physics

definition

Relation Between Angular Velocity and Linear Velocity

Relation Between Angular Velocity and Linear Velocity:

v = w r

definition

Relation between frequency, time period and angular velocity

w = \frac{2 π}{T} = 2 π ν

rad/s

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Angular Velocity in terms of linear velocity

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Angular Velocity in terms of Linear Velocity

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Important Questions

Obtain the relation between linear velocity and angular velocity

Medium

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Angular velocity of minute hand of a clock is

Easy

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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 :

Medium

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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?

Medium

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

Medium

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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?

Medium

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

Easy

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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 $B (∠ A O B = 4 0^{\circ}) ?$

Medium

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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.

Medium

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The angular velocity of a wheel is $70 r a d / s$ . If the radius of the wheel is $0.5 m$ , then linear velocity of the wheel is

Easy

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definition

Relation Between Angular Velocity and Linear Velocity

definition

Relation between frequency, time period and angular 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

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(∠AOB=40∘)?

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

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 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 $B (∠ A O B = 4 0^{\circ}) ?$

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 angular velocity of a wheel is $70 r a d / s$ . If the radius of the wheel is $0.5 m$ , then linear velocity of the wheel is