First slide

Simple harmonic motion

Question

Two simple harmonic oscillators with amplitudes in the ratio 1:2 are having the same total energy, the ratio of their frequencies is

Moderate

A

1:4
B

1:2
C

2:1
D

4:1

Solution

$i f a m p l i t u d e A_{1} = A t h e n A_{2} = 2 A t o t a l e n e r g y o f b o t h o s c i l l a t o r s i s s a m e t o t a l e n e r g y = \frac{1}{2} m w^{2} A^{2} \frac{1}{2} m {w_{1}}^{2} {A_{1}}^{2} = \frac{1}{2} m {w_{2}}^{2} {A_{2}}^{2} \Rightarrow \frac{1}{2} m {w_{1}}^{2} A^{2} = \frac{1}{2} m {w_{2}}^{2} 2^{2} A^{2} \Rightarrow \frac{{w^{2}}_{1}}{{w_{2}}^{2}} = 4 t a k e s q u a r e r o o t \Rightarrow \frac{w_{1}}{w_{2}} = 2 \Rightarrow \frac{2 π f_{1}}{2 π f_{2}} = 2 \Rightarrow \frac{f_{1}}{f_{2}} = 2 \$

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