Problem 39 - University Physics with Modern Physics

Question

A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the antinodes of 0.300 cm. (a) What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire. (c) Find the maximum transverse velocity and acceleration of particles in the wire.

Step-by-Step Solution

Answer

(a) 96.0 m/s, (b) 460.8 N, (c) 1.13 m/s, 426.2 m/s².

Step 1: Identify Formula for Wave Speed

The formula for the speed of a wave on a stretched string is given by \( v = f \cdot \lambda \), where \( v \) is the wave speed, \( f \) is the frequency, and \( \lambda \) is the wavelength. In the fundamental mode of vibration, the wavelength is twice the distance between the fixed ends, so \( \lambda = 2L = 2 \times 0.8 \text{ m} = 1.6 \text{ m} \).

Step 2: Calculate the Speed of the Wave

Substitute the given frequency and the calculated wavelength into the wave speed formula: \( v = 60.0 \text{ Hz} \times 1.6 \text{ m} = 96.0 \text{ m/s} \). Therefore, the speed of propagation of transverse waves in the wire is \( 96.0 \text{ m/s} \).

Step 3: Determine Formula for Tension in the Wire

The wave speed is also related to the tension \( T \) and the linear mass density \( \mu \) by the formula \( v = \sqrt{\frac{T}{\mu}} \). The linear mass density \( \mu \) can be calculated from the total mass \( M \) and the length \( L \): \( \mu = \frac{M}{L} = \frac{0.040 \text{ kg}}{0.8 \text{ m}} = 0.05 \text{ kg/m} \).

Step 4: Calculate the Tension in the Wire

Rearrange the formula \( v = \sqrt{\frac{T}{\mu}} \) to solve for tension: \( T = v^2 \cdot \mu \). Substitute \( v = 96.0 \text{ m/s} \) and \( \mu = 0.05 \text{ kg/m} \) into the equation: \( T = (96.0)^2 \cdot 0.05 = 460.8 \text{ N} \). Thus, the tension in the wire is \( 460.8 \text{ N}\).

Step 5: Compute Maximum Transverse Velocity

The maximum transverse velocity \( v_{max} \) of a particle in SHM is given by \( v_{max} = \omega A \), where \( \omega = 2\pi f \) is the angular frequency and \( A \) is the amplitude. First calculate \( \omega = 2\pi \times 60.0 \text{ Hz} \approx 376.99 \text{ rad/s} \) and use amplitude \( A = 0.003 \text{ m} \). Thus, \( v_{max} = 376.99 \times 0.003 \text{ m} = 1.13 \text{ m/s} \).

Step 6: Calculate Maximum Transverse Acceleration

The maximum transverse acceleration \( a_{max} \) of a particle is given by \( a_{max} = \omega^2 A \). Using \( \omega = 376.99 \text{ rad/s} \) and \( A = 0.003 \text{ m} \), \( a_{max} = (376.99)^2 \times 0.003 \approx 426.2 \text{ m/s}^2 \).

Key Concepts

Transverse Waves

Transverse waves are a type of wave where the displacement of the medium is perpendicular to the direction of wave propagation. These are commonly observed in strings, water surfaces, and electromagnetic waves. In the context of a stretched wire, as described in the exercise, transverse waves move from one end of the wire to the other while causing the wire particles to oscillate up and down. This perpendicular movement is what characterizes them as transverse.

• Perpendicular Oscillation: The motion is perpendicular to the direction of energy transfer.
• Wave Propagation: Energy moves along the wave while the medium does not move with it.

Understanding the transverse nature is crucial to analyzing wave speed and tension in the wire, as these properties can affect how fast the waves travel and the energy they carry.

Wave Speed

Wave speed is a fundamental concept in wave motion, indicating how fast a wave propagates through a medium. In a string or wire, as given in the problem, wave speed depends on the frequency and wavelength. The formula to find the wave speed is:\[ v = f \cdot \lambda \]where \( f \) is the frequency and \( \lambda \) is the wavelength. In our exercise:

• The frequency \( f \) is 60.0 Hz.
• The wavelength \( \lambda \) is twice the length of the wire segment, calculated as 1.6 meters.
• Therefore, the wave speed \( v \) is 96.0 m/s.

Using wave speed, we can determine how quickly energy or a signal can travel along the wire.

Tension in the Wire

Tension is a force exerted along the string or wire, which in wave mechanics, is crucial for determining wave speed. The relationship between tension \( T \), wave speed \( v \), and the linear mass density \( \mu \) of the wire is expressed by:\[ v = \sqrt{\frac{T}{\mu}} \]To find the tension in our scenario, we need to calculate:

• Linear Mass Density: This is the mass per unit length, calculated as 0.05 kg/m from the given mass and length of the wire.
• Rearranging the Formula: \( T = v^2 \cdot \mu \)
• The calculated tension is 460.8 N.

Higher tension results in higher wave speed, showing how physical forces impact wave properties.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a form of periodic motion where the restoring force is directly proportional to displacement, typically modeled by the sine or cosine function. In wave motion, particles of a medium like a wire moving transversely exhibit SHM:The maximum transverse velocity and acceleration of particles can be calculated using the following formulas:

• Maximum Transverse Velocity \( v_{max} \): \( v_{max} = \omega A \), where \( \omega \) is the angular frequency \( 2\pi f \) and \( A \) is the amplitude.
• With the given frequency and amplitude, \( v_{max} \) comes out to be 1.13 m/s.
• Maximum Transverse Acceleration \( a_{max} \): \( a_{max} = \omega^2 A \).
• Using the same variables, \( a_{max} \) is calculated as 426.2 \( m/s^2 \).

Understanding SHM helps in grasping how particles of the wire move, providing insights into the motion and energy distribution in the wave.