A simple harmonic oscillator with a period of 2.0s is subject to damping so that it loses one percent of its amplitude per cycle. About how much energy does this oscillator lose per cycle?

Options

A. 0.005
B. 0.01
C. 0.02
D. 0.03

Answer

To solve the problem of how much energy a damped simple harmonic oscillator loses per cycle, we can follow these steps:

### Step 1: Understand the relationship between energy and amplitude
The total energy \( E \) of a simple harmonic oscillator is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the spring constant and \( A \) is the amplitude of the oscillation.

### Step 2: Determine the change in amplitude
We are told that the oscillator loses 1% of its amplitude per cycle. This means that if the initial amplitude is \( A \), the amplitude after one cycle becomes:
\[ A' = A - 0.01A = 0.99A \]

### Step 3: Calculate the initial energy and the energy after one cycle
The initial energy \( E \) is:
\[ E = \frac{1}{2} k A^2 \]
The energy after one cycle, when the amplitude has decreased to \( 0.99A \), is:
\[ E' = \frac{1}{2} k (0.99A)^2 = \frac{1}{2} k (0.9801A^2) \]

### Step 4: Find the energy lost per cycle
The energy lost \( \Delta E \) per cycle is the difference between the initial energy and the energy after one cycle:
\[ \Delta E = E - E' = \frac{1}{2} k A^2 - \frac{1}{2} k (0.9801A^2) \]
\[ \Delta E = \frac{1}{2} k A^2 (1 - 0.9801) = \frac{1}{2} k A^2 (0.0199) \]

### Step 5: Express the energy lost in terms of the initial energy
We can express the energy lost as a fraction of the initial energy:
\[ \Delta E = 0.0199 \cdot \frac{1}{2} k A^2 = 0.0199 \cdot E \]

### Step 6: Finalize the answer
Since we are interested in the energy lost per cycle, we can conclude that the oscillator loses approximately 1.99% of its total energy per cycle. Given that this is close to 2%, we can say:
\[ \Delta E \approx 0.02 E \]

Thus, the energy lost per cycle is about 2% of the initial energy.

### Summary
The oscillator loses approximately 2% of its energy per cycle due to damping.

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