The air pressure variations in a sound wave cause the eardrum to vibrate.

Find the maximum velocity of the eardrum for vibrations of amplitude 5.46 × 10−8 m at a frequency of 20.0 Hz.

m/s

To find the maximum velocity of the eardrum, we can use the relationship between amplitude, frequency, and the maximum velocity in simple harmonic motion (SHM). The formula for the maximum velocity \( v_{\text{max}} \) in SHM is:

\[
v_{\text{max}} = 2 \pi f A
\]

where:
- \( f \) is the frequency,
- \( A \) is the amplitude of vibration.

### Given values:
- \( A = 5.46 \times 10^{-8} \, \text{m} \)
- \( f = 20.0 \, \text{Hz} \)

### Solution
1. Substitute the given values into the formula:

   \[
   v_{\text{max}} = 2 \pi \times 20.0 \times 5.46 \times 10^{-8}
   \]

2. Calculating the result:

   \[
   v_{\text{max}} = 2 \pi \times 20.0 \times 5.46 \times 10^{-8} \approx 6.86 \times 10^{-6} \, \text{m/s}
   \]

### Answer
The maximum velocity of the eardrum is approximately:
\[
\boxed{6.86 \times 10^{-6} \, \text{m/s}}
\]

Would you like further details, or have any questions?

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Here are 5 additional questions for practice:
1. How would the maximum velocity change if the frequency were doubled?
2. What is the relationship between amplitude and maximum acceleration in SHM?
3. If the amplitude were halved, how would it affect the maximum velocity?
4. How would a higher frequency affect the energy of the eardrum's motion?
5. What formula would you use to find the maximum displacement in SHM?

**Tip:** In simple harmonic motion, frequency and amplitude are the main factors influencing maximum velocity.

The air pressure variations in a sound wave cause the eardrum to vibrate. Find the maximum velocity of the eardrum for vibrations of amplitude 5.46 × 10−8 m at a frequency of 20.0 Hz.

Discover how to calculate the maximum velocity of an eardrum vibrating due to sound waves, given an amplitude of 5.46 × 10^-8 m and a frequency of 20.0 Hz. This problem covers key concepts in simple harmonic motion, suitable for students in Grades 11-12.

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