First slide

Simple harmonic motion

Question

A particle undergoing simple harmonic motion has time dependent displacement given by $x (t) = A \sin \frac{π t}{90}$ . The ratio of kinetic to potential energy of this particle at time $t = 30 s$ will be

Moderate

A

1/3
B

2/3
C

1/9
D

1/6

Solution

$l e t d i s p l a c e m e n t e q u a t i o n o f a n o s c i l l a t o r b e y = A \sin ω t, h e r e ω i s a n g u l a r v e l o c i t y, A i s A m p l i t u d e d i f f e r e n t i a t e y w i t h r e s p e c t t o t i m e t, t o g e t v e l o c i t y v = \frac{d y}{d t} = A (\cos ω t) ω K E = \frac{1}{2} m v^{2} P E = \frac{1}{2} m ω^{2} y^{2} S u b s t i t u t e d i s p l a c e m e n t a n d v e l o c i t y i n P E, K E a n d t a k e r a t i o, \frac{K E}{P E} = \frac{\frac{1}{2} m ω^{2} A^{2} \cos^{2} ω t}{\frac{1}{2} m ω^{2} A^{2} \sin^{2} ω t} \frac{K E}{P E} = \cot^{2} ω t, b y t h e q u e s t i o n w = \frac{π}{90} (at time t = 30) \Rightarrow \frac{K E}{P E} = \cot^{2} (\frac{π}{90} \times 30) = \cot^{2} (\frac{π}{3}) = \frac{1}{3}$

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