A particle is moving in a circular orbit with a constant tangential acceleration. After certain time T has elapsed after the beginning of Motion from rest,the angle between the total acceleration and the direction along the radius becomes equal to 45 degree. what is angular displacement of the particle at this instant

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Solution

$T h e \tan g e n t i a l a c c e l e r a t i o n o f t h e p a r t i c l e a = c o n s \tan t (c) t o t a l a c c e l e r a t i o n A = \sqrt{(c^{2} + α^{2})} g i v e n \sqrt{(c^{2} + α^{2})} \cos 45 ° = α o r \sqrt{(c^{2} + α^{2})} = \sqrt{2} α o r c = α F u r t h e r \frac{d ω}{d t} = c o r d ω = c d t o r ω = c t + k n o w w h e n t = 0, ω = 0 s o k = 0, t h e r e f o r e, ω = c t o r \frac{d θ}{d t} = c t o r d θ = c t d t o r θ = \frac{c t^{2}}{2} + k' a g a i n w h e n t = 0, θ = 0 s o k' = 0 . T h e r e f o r e a n g u l a r d i s p l a c e m e n t θ = \frac{c t^{2}}{2} .$

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