Problem 22 - University Physics with Modern Physics

Question

A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wave with frequency 120.0 Hz and amplitude 1.6 mm travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

Step-by-Step Solution

Answer

(a) 0.928 W; (b) 0.232 W when amplitude is halved.

Step 1: Calculate the wave speed

First, determine the speed of the wave on the wire using the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T = 25.0 \) N is the tension and \( \mu = \frac{m}{L} \) is the linear mass density. Here, \( m = 3.00 \, \text{g} = 0.0030 \, \text{kg} \) and \( L = 80.0 \, \text{cm} = 0.80 \, \text{m} \). First, calculate \( \mu = \frac{0.0030}{0.80} = 0.00375 \, \text{kg/m} \). Substitute \( \mu \) into the wave speed formula: \( v = \sqrt{\frac{25}{0.00375}} = \sqrt{6666.67} \approx 81.65 \, \text{m/s} \).

Step 2: Calculate the average power of the wave

The average power \( P \) carried by a wave on a stretched string can be calculated using the formula \( P = \frac{1}{2} \mu v \omega^2 A^2 \), where \( \omega \) is the angular frequency and \( A \) is the amplitude. First, find \( \omega = 2\pi f = 2\pi (120) = 240\pi \, \text{rad/s} \). Convert amplitude \( A \) to meters: \( A = 1.6 \, \text{mm} = 0.0016 \, \text{m} \). Substituting all values: \[ P = \frac{1}{2} \times 0.00375 \times 81.65 \times (240\pi)^2 \times (0.0016)^2 \approx 0.928 \, \text{W} \].

Step 3: Analyze effect of halving the amplitude

If the wave amplitude is halved, \( A \) becomes \( 0.0008 \, \text{m} \). Since power \( P \) is proportional to \( A^2 \), the new power can be calculated as follows: Halving the amplitude means the new power \( P' \) is \( (0.0008/0.0016)^2 \times 0.928 \, \text{W} = \frac{1}{4} \times 0.928 \, \text{W} = 0.232 \, \text{W} \).

Step 4: Conclusion: Final Answer

The average power carried by the wave is approximately 0.928 W. If the amplitude is halved, the average power becomes 0.232 W.

Key Concepts

wave speed

Wave speed is an essential concept in understanding waves on a string. It refers to how fast a wave travels along the string, which depends largely on the tension in the string and the linear mass density. The formula for the wave speed (\(v\)) on a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
Here, \(T\) is the tension in the string in newtons (N), and \(\mu\) is the linear mass density in kilograms per meter (kg/m).

• The higher the tension, the faster the wave travels.
• Conversely, a larger mass density, meaning a heavier string, results in a slower wave speed.

In practical terms, knowing the wave speed helps us understand how quickly the energy is being transferred along the string, which is crucial for calculating wave power.

tension in a string

The tension in a string is the force exerted along the string that keeps it stretched. Tension is a key factor that impacts both wave speed and the overall dynamics of the wave. In the given problem, the piano wire is stretched with a tension of 25.0 N. This force is necessary to maintain the wire at a certain tightness. Tension affects:

• Wave Speed: As previously mentioned, higher tension means higher wave speed.
• Wave Frequency: While more indirectly, tension can also influence the frequency at which the string vibrates.

Tension is crucial in musical instruments like the piano since it affects the note played—the tighter the string, the higher the pitch.

linear mass density

Linear mass density (\(\mu\)) is a measure that combines the mass and length of the string into a single value. It indicates how much mass is distributed along each unit length of the string. For the piano wire in the example, with a mass of 3.00 g and a length of 80.0 cm, the linear mass density is calculated as:\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{0.0030 \, \text{kg}}{0.80 \, \text{m}} = 0.00375 \, \text{kg/m} \]Understanding linear mass density is vital for two main reasons:

• It directly affects the wave speed; a higher \(\mu\) means a slower wave.
• It is essential for determining the wave's transport of energy, as seen in the power formula.

When designing strings for specific purposes like music or communication cables, considering \(\mu\) ensures the desired performance.

wave amplitude

Wave amplitude is the maximum displacement from the rest position that the particles of the medium experience as the wave passes. The exercise provides an amplitude of 1.6 mm (or 0.0016 m) for the wave traveling along the piano wire. Amplitude plays a significant role in determining the power that a wave transfers:

• The wave's energy is directly proportional to the square of its amplitude, which is why halving the amplitude to 0.0008 m results in the power being reduced to a quarter.
• Simply put, a larger amplitude means a more energetic wave, leading to higher power.

Understanding how amplitude affects power can help in applications where controlling the energy transfer is crucial. For instance, in musical contexts, a larger amplitude typically means a louder sound.