At a given point of time the value of displacement of a simple harmonic oscillator is given as . If amplitude is 40 cm and kinetic energy at that time is 200 J, the value of force constant is . The value of is __________ . [2023]

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Solution

Q.
At a given point of time the value of displacement of a simple harmonic oscillator is given as $y = A \cos (30 °)$ . If amplitude is 40 cm and kinetic energy at that time is 200 J, the value of force constant is $1.0 \times 10^{x} {Nm}^{- 1}$ . The value of $x$ is __________ . [2023]

Ans.
(4)

$General equation for displacement is given by$

$x = A \sin (ω t + ϕ)$

$At given time \Rightarrow ω t + ϕ = 60 °$

$\Rightarrow x = 40 \times \frac{\sqrt{3}}{2} = 20 \sqrt{3} cm$

$\Rightarrow A = 40 cm$

$\Rightarrow K.E. = \frac{1}{2} k (A^{2} - x^{2}) = 200$

$200 = \frac{1}{2} k (\frac{1600 - 1200}{100 \times 100})$

$400 \times 100 \times 100 = k \times 400$

$\Rightarrow k = 10^{4}$

$\Rightarrow x = 4$

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