Question

prove that in circular motion a = r alpha​

Answer 1

Question :-

Prove that in circular motion " a = r.α"

Let's prove :-

Let a particle is moving on a circular path with uniform motion then We know that for it $\bold{v = r. \omega}$ ---(1)

if the velocity of the particle changes with time then differentiate equation (1) with respect to

$\: \: \: \: \: \: \: \: \rightarrow \rm{\frac{dv}{dt} } = \frac{d(r \times \omega)}{dt} \: \:\: \: \: \: \: \: \\ \: \: \: \: \: \: \: \: \: \implies\rm{ \frac{dv}{dt} } = r. \frac{dw}{dt} - - (2)\: \: \: \: \: \:$

here,

$\rm{ \large{\frac{dv}{dt} }} = \tiny{ \frac{ change \: in \: velo.}{time \: taken }} = \normalsize{linear \: acc. = \green{a}} \\ \rm{ \large{ \frac{d \omega}{dt} } = \tiny{ \frac{change \: in \: angular \: velo.}{time} = }} \: \normalsize{angular \: acc. = \green{ \alpha}}$

these values are putting in equation $2^{nd}$ , .

then, we get

$\: \: \: \: \: \: : \implies \boxed{\huge{ \green{a = r. \alpha}}} \: \:$

Hence proved !!

#Gunjan