JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Elasticity & SHM

The general displacement of a simple harmonic oscillator is

x = A sin ω t

. Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when

t = \frac{T}{β}

. The value of

β

is ______.

Answer: 8

Solution

The stored energy is

U = \frac{1}{2} k x^{2} = \frac{1}{2} k A^{2} sin^{2} ω t

, so its time slope is

\frac{d U}{d t} = k A^{2} ω sin ω t cos ω t = \frac{k A ^{2} ω}{2} sin 2 ω t

. This slope first reaches its greatest value when

sin 2 ω t = 1

, that is

2 ω t = \frac{π}{2}

, giving

t = \frac{π}{4 ω} = \frac{T}{8}

. Hence

β = 8

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Integer answer