The amplitude of a damped oscillator decreases to 0.9 times its origin

1.	The amplitude of a damped oscillator decreases to 0.9 times its original magnitude in 5s. In another 10 s it will decrease to α times its original magnitude, then α is equal to :(a) 0.7 (b) 0.81 (c) 0.729 (d) 0.6
Answer» Answer : (c) 0.729 The amplitude of damped oscillator is, A = A₀e^-kt, Where k = \(\frac{b}{2m}\) At t = 5s, 0.9 A₀ = A₀e^-5k ⇒ e^-5k= 0.9 At t = 15s, αA₀ = A₀e^-15k ⇒ e^-15k= α ⇒ e⁽^-5k)3= α ⇒ α = (0.9)³ = 0.729

The amplitude of a damped oscillator decreases to 0.9 times its original magnitude in 5s. In another 10 s it will decrease to α times its original magnitude, then α is equal to :(a) 0.7 (b) 0.81 (c) 0.729 (d) 0.6

Answer»

Answer : (c) 0.729

The amplitude of damped oscillator is,

A = A₀e^-kt,

Where k = \(\frac{b}{2m}\)

At t = 5s,

0.9 A₀ = A₀e^-5k

⇒ e^-5k= 0.9

At t = 15s,

αA₀ = A₀e^-15k

⇒ e^-15k= α

⇒ e⁽^-5k)3= α

⇒ α = (0.9)³

= 0.729

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The amplitude of a damped oscillator decreases to 0.9 times its original magnitude in 5s. In another 10 s it will decrease to α times its original magnitude, then α is equal to :(a) 0.7 (b) 0.81 (c) 0.729 (d) 0.6

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