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Physics Question on Oscillations

If the ratio of the mechanical energy of a mass whose oscillations are damped to that of its free oscillations is e−3t/me^{-3 t/m} , then the ratio of the amplitudes of the damped oscillations to that of free oscillations is

A

e−3tme ^{-\frac{3t}{m}}

B

e−3t2me ^{-\frac{3t}{2m}}

C

e−6tme ^{-\frac{6t}{m}}

D

e−3t2me ^{-\frac{3t}{\sqrt{2 }m}}

Answer

e−3t2me ^{-\frac{3t}{2m}}

Explanation

Solution

If EE and E0E_{0} represent the mechanical energy of the damped and free oscillations respectively, then as per question
EE0=e−3t/m\frac{E}{E_{0}}=e^{-3t/m}
As mechanical energy ∝(amplitude)2\propto {\text(amplitude)}^{2}
∴\therefore The ratio of the amplitudes of the damped oscillations to that of free oscillations is
AA0=(EE0)1/2\frac{A}{A_{0}}=\left(\frac{E}{E_{0}}\right)^{1 /2}
=(e−3t/m)1/2=\left(e^{-3t /m}\right)^{1 /2}
=e−3t/2m=e^{-3t /2m}