Question

. What do you mean by uniform circular motion? Derive an expression for acceleration of an
object in uniform circular motion.

Answer 1

$$\begin{gathered} Whenabodymovesinacircularpathwithcons\tan tangularspeedorwithauniformspeed, \\ themotionofthebodyissiadtobecircularmotion. \\ Letusconsiderabodymovinginacircularpathinsuchawaythatitsdisplacementis \\ governedby: \\ x=r\sin \omega t \\ whentimetakenis\frac{T}{4}andpathdifference=\frac{\pi }{2}thenx\left(r,\frac{\pi }{2}\right)=r\sin \left(\omega t+\frac{\pi }{2}\right)=r\cos \omega t \\ andx\left(r,\pi \right)=-rsin\omega t \\ andagainatx\left(r,2\pi \right)=rsin\left(\omega t+2\pi \right)=rsin\omega t \\ Sothemotioniscircular. \\ Nowthevelocityofthebodywithtimeisgivenby: \\ v=\frac{dx}{dt}=\omega r\cos \omega t \\ andtheaccelerationisgivenby: \\ a=\frac{d^{2}x}{dt}=-{\omega }^{2}r\sin \omega t=-{\omega }^{2}x \\ Thereforeaccelerationa=-{\omega }^{2}xforabodymovingalongacircularpath. \end{gathered}$$