Question

can anyone please explain me how a.w.cos.w.t came????

(SHM) ossiclation chapter of class 11​

Answer 1

Question :-

How did this came ?

$\to \bf if \: \sf \: x = \: a \sin( \omega \: t) \\ \\ \to \sf \: v = a \omega \cos( \omega \: t)$

Explanation :-

$\bf \: here \\ \\ \to \sf v = velocity \: \\ \to \sf \omega = angular \: frequency \\ \to \sf \: x = displacement \\ \to\sf a = amplitude \:$

We have ,

$\to \sf x = \: a \sin ( \omega \: t) \\ \\ \sf \: differentiate \: with \: respect \: to \: t \: \\ \\ \to \sf \frac{dx}{dt} = \dfrac{d}{dt} a \sin ( \omega \: t) \\ \\ \: \: \: \: \: \: \: \: \: \: \: \boxed{ \because \sf \frac{d}{dx} \sin x = \cos x} \\ \therefore \\ \\ \to \sf \frac{dx}{dt} = a \cos( \omega \: t) \frac{d}{dt} \omega t \\ \: \\ \sf \red{ here \: \: a \: and \: } \green{ \omega \: are \: constant \: } \\ \\ \: \: \: \: \: \: \: \: \boxed{ \sf \because \frac{d}{dt} x = 1} \\ \therefore$

$\to \sf \: \frac{dx}{dt} = a \: \omega \: t \cos ( \omega \: t) \\ \\ \boxed{ \because \sf \frac{dx}{dt} = v \: } \\ \\ \to \red{ \boxed{ \sf \green{v = a \: \omega \cos( \omega \: t)}}}$