Question

relation between linear and angular velocity ​

Answer 1

${\mathcal{\red{ANSWER}}}$

Linear velocity is defined as the rate of change of displacement. The unit of linear velocity is m/s.

Angular velocity of a body is the rate of change of angular displacement over a period of time. The unit of angular velocity is rad/s.

Relation between Linear velocity and angular velocity:

Note: See figure in attachment file.

$\sf{Let\; \omega \rightarrow Uniform\;angular\;velocity\;of\;point\;object\;moving \;along \;PQ}$

$\sf{v \rightarrow Linear\;speed}$

$\sf{r \rightarrow Radius\;of\;circular\;path}$

$\sf{t \rightarrow Time \;at \;which\; the \;object\; is \;at \;point \;P}$

$\sf{t+\Delta t \rightarrow Time\;at\;which\;the\;object\;is\;at\;Q}$

$\sf{Let\; \angle POQ = \Delta \theta}$

And,

$\sf{\overrightarrow{OQ}=\overrightarrow{r}+\overrightarrow{\Delta r}}$

It means that an object describes an arc PQ of length Δl in time interval Δt.

$\sf{\therefore v = \dfrac{\Delta l}{\Delta t}}$

$\sf{\implies \Delta l = v\Delta t\;\;\;...........(1)}$

$\sf{\omega = \dfrac{\Delta \theta}{\Delta t}}$

$\sf{\implies \Delta \theta = \omega \Delta t \;\;\;\;\;.........(2)}$

We know,

$\sf{\Delta \theta = \dfrac{\Delta l}{r} \;\;\;\;\;\;\;\;\Bigg[\therefore Angle = \dfrac{Arc}{Radius}\Bigg]}$

From equations (1) and (2),

$\sf{\omega \Delta t=\dfrac{v\Delta t}{r}}$

${\boxed{\boxed{\bf{\therefore v = r\omega}}}}$

Direction of velocity at any point in circular motion is directed along the tangent to the circle at that point in the direction of motion.

Answer 2

Answer:

Answer:

Heya.....!!!

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Given in the question :-

( v ) => Linear Velocity .

( ω ) => Angular Velocity .

There is a realtion between linear displacement and angular displacement .

let angular displacement be ( x )

Angle which is moved by particle (  θ )

radius ( r )

=>  θ = x / r

=> x = r θ .............( i )

in this equation ( i ) divide both side by ( t ) time

=> x / t = rθ/t

=> x / t. = v ,,  θ/t. = ω

v = rω

Hence the realtion between angular velocity and linear velocity is

➡ ♦ v = r × ω ♦ .

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