Question

In a time ‘’t", the amplitude of vibrations of a damped oscillator becomes half of its initial value, then the mechanical energy of the oscillator decreases by:

• A. 40%
• B. 20%
• C. 75%
• D. 50%

Hint:

The energy of a damped oscillator decreases exponentially over time, and when the amplitude is halved, the energy reduces to one-fourth of its initial value.

Answer 1

The Correct Option is C

The mechanical energy of a damped oscillator is proportional to the square of its amplitude: \[ E \propto A^2 \] Given that the amplitude becomes half of its initial value: \[ A' = \frac{A}{2} \] The new mechanical energy is: \[ E' = k \left(\frac{A}{2}\right)^2 = \frac{1}{4} kA^2 \] The percentage decrease in energy is: \[ \frac{E - E'}{E} \times 100 = \frac{kA^2 - \frac{1}{4}kA^2}{kA^2} \times 100 \] \[ = \left(1 - \frac{1}{4}\right) \times 100 = 75\% \]