Question:
If the amplitude of a lightly damped oscillator decreases by \(1.5\%\), then the mechanical energy of the oscillator lost in each cycle is
If the amplitude of a lightly damped oscillator decreases by \(1.5\%\), then the mechanical energy of the oscillator lost in each cycle is
Show Hint
In SHM,
\[
E\propto A^2
\]
So, a small percentage decrease in amplitude produces approximately double the percentage decrease in energy.
Updated On: Jun 22, 2026
- \(1.5\%\)
- \(0.75\%\)
- \(6\%\)
- \(3\%\)
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The Correct Option is D
Solution and Explanation
Step 1: Relation between energy and amplitude.
For a simple harmonic oscillator, \[ E\propto A^2 \] Thus, \[ \frac{E_2}{E_1}=\left(\frac{A_2}{A_1}\right)^2 \]
Step 2: Calculate the new amplitude.
Amplitude decreases by \[ 1.5\% \] Hence, \[ A_2=98.5\% \text{ of } A_1 \] \[ A_2=0.985A_1 \]
Step 3: Find the ratio of energies.
Using \[ E\propto A^2, \] \[ \frac{E_2}{E_1}=(0.985)^2 \] \[ \frac{E_2}{E_1}=0.970225 \] Therefore, the fraction of energy lost is \[ 1-0.970225 \] \[ =0.029775 \] \[ \approx 0.03 \] Hence, percentage loss in energy is \[ 0.03\times 100=3\% \]
Step 4: Final conclusion.
Therefore, the mechanical energy lost in each cycle is \[ \boxed{3\%} \]
For a simple harmonic oscillator, \[ E\propto A^2 \] Thus, \[ \frac{E_2}{E_1}=\left(\frac{A_2}{A_1}\right)^2 \]
Step 2: Calculate the new amplitude.
Amplitude decreases by \[ 1.5\% \] Hence, \[ A_2=98.5\% \text{ of } A_1 \] \[ A_2=0.985A_1 \]
Step 3: Find the ratio of energies.
Using \[ E\propto A^2, \] \[ \frac{E_2}{E_1}=(0.985)^2 \] \[ \frac{E_2}{E_1}=0.970225 \] Therefore, the fraction of energy lost is \[ 1-0.970225 \] \[ =0.029775 \] \[ \approx 0.03 \] Hence, percentage loss in energy is \[ 0.03\times 100=3\% \]
Step 4: Final conclusion.
Therefore, the mechanical energy lost in each cycle is \[ \boxed{3\%} \]
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