The amplitude of a lightly damped oscillator decrease by 3% during each cycle. What fraction of the energy of the oscillator is lost in each full cycle?

To determine the fraction of energy lost in each full cycle of a lightly damped oscillator, we can start by understanding how the energy relates to amplitude. When an oscillator loses energy due to damping, it is primarily the amplitude that decreases, and since the energy stored in the oscillator is proportional to the square of the amplitude, we can derive the fraction of energy lost from the change in amplitude.

Understanding the Energy Relationship

The energy \( E \) of an oscillator is given by the equation:

E = \frac{1}{2} k A^2

where \( k \) is the spring constant and \( A \) is the amplitude. If the amplitude decreases by 3% during each cycle, we can express this mathematically:

• Let the initial amplitude be \( A \).
• After one cycle, the new amplitude \( A' \) will be:

A' = A (1 - 0.03) = 0.97A

Calculating the Energies

Now, let’s calculate the energy before and after one cycle:

• The initial energy \( E \) is:

E = \frac{1}{2} k A^2

• The energy after one cycle \( E' \) is:

E' = \frac{1}{2} k (0.97A)^2 = \frac{1}{2} k (0.9409 A^2)

Finding the Fraction of Energy Lost

To find the fraction of energy lost, we subtract the final energy from the initial energy:

Energy lost = E - E' = \frac{1}{2} k A^2 - \frac{1}{2} k (0.9409 A^2)

This simplifies to:

Energy lost = \frac{1}{2} k A^2 (1 - 0.9409) = \frac{1}{2} k A^2 (0.0591)

Now, we can find the fraction of energy lost:

Fraction lost = \frac{Energy lost}{Initial Energy} = \frac{\frac{1}{2} k A^2 (0.0591)}{\frac{1}{2} k A^2} = 0.0591

Final Result

Thus, the fraction of the energy lost in each full cycle of the lightly damped oscillator is approximately 0.0591, or about 5.91%. This means that with each cycle, the oscillator loses roughly 5.91% of its total energy due to damping effects.