Name: Kinetic Energy of Rotational Motion
Uploaded: 2019-07-19
Duration: 9 min 3 s
Description: We all know the formula for translational kinetic energy, but we need rotational kinetic energy for a rotating object. Let's learn the expression for it.

Question

Establish relations between angular acceleration and linear acceleration. Find an expression for the centripetal acceleration.

Accepted Answer

Angular Acceleration - It is the rate of change of angular velocity with a time of an object in motion. The angular acceleration is also known as rotational acceleration. The vector direction of the acceleration is perpendicular to the plane where the rotation takes place.

We know that,

Linear acceleration is equal to the rate of change of linear velocity i.e.

$\Rightarrow a = \frac{d v}{d t} . . . . . . (i)$

Similarly, angular acceleration is equal to the rate of change of angular velocity i.e.

$\Rightarrow α = \frac{d ω}{d t} . . . . . . (i i)$

From the first and second equation,

$\frac{a}{α} = \frac{d v}{d ω} = \frac{d r ω}{d t} = r \frac{d ω}{d t}$

As, $v = ω r$ and $r$ is constant.

$a = α . r$

Now Centripetal Acceleration,

Angular velocity, $ω = \frac{∆ θ}{∆ t}$

Let $\vec{v_{1}}, \vec{v_{2}}$ be the velocity vectors.

$ω ∆ t = \frac{|∆ \vec{v}|}{|\vec{v}|}$

We know that, $v = ω r$

$\frac{|∆ \vec{v}|}{∆ t} = |\vec{v}| ω = (ω r) ω = ω^{2} r$

when $∆ t \to 0$ then $\frac{|∆ \vec{v}|}{∆ t}$ represents the magnitude of the centripetal acceleration at point $P$ which is given by,

$|\vec{a}| = \frac{|∆ \vec{v}|}{∆ t} = ω^{2} r = {(\frac{v}{r})}^{2} \Rightarrow |\vec{a}| = ω^{2} r$

Establish relations between angular acceleration and linear acceleration. Find an expression for the centripetal acceleration.

Important Questions on Motion of System of Particles and Rigid Body

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