Problem 19 - University Physics with Modern Physics

Question

A thin, 75.0 -cm wire has a mass of 16.5 \(\mathrm{g} .\) One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in newtons) must you adjust the screw so that a transverse wave of wavelength 3.33 \(\mathrm{cm}\) makes 875 vibrations per second? (b) How fast would this wave travel?

Step-by-Step Solution

Answer

(a) Adjust tension to 18.67 N. (b) Wave speed is 29.14 m/s.

Step 1: Convert Units

Firstly, convert the units to the standard metric system. The length of the wire, \(75.0\, \text{cm}\), is converted to meters: \(0.75\, \text{m}\). The mass of the wire, \(16.5\, \text{g}\), is converted to kilograms: \(0.0165\, \text{kg}\). The wavelength \(3.33\, \text{cm}\) is converted to meters: \(0.0333\, \text{m}\).

Step 2: Determine the Wave Velocity

Use the formula for the velocity of a wave, \( v = f \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength. The frequency is \(875\, \text{Hz}\) and the wavelength is \(0.0333\, \text{m}\), so the velocity \( v \) is \( v = 875 \times 0.0333 \). Calculating this gives \( v = 29.1375\, \text{m/s}\).

Step 3: Calculate Linear Mass Density

The linear mass density \( \mu \) of the wire is \( \mu = \frac{m}{L} \), where \( m \) is the mass and \( L \) is the length of the wire. Substituting the known values, \( \mu = \frac{0.0165}{0.75} = 0.022\, \text{kg/m}\).

Step 4: Calculate Tension in the Wire

Using the wave velocity and linear mass density, calculate the tension \( T \) in the wire using the formula \( v = \sqrt{\frac{T}{\mu}} \). Rearrange to solve for \( T \): \( T = v^2 \mu \). Substituting the values, \( T = (29.1375)^2 \times 0.022\). Calculating gives \( T \approx 18.67\, \text{N}\).

Step 5: Verify Wave Speed Calculation

Use the derived tension to verify the speed of the wave using \( v = \sqrt{\frac{T}{\mu}} \). Substituting \( T = 18.67 \) and \( \mu = 0.022 \), verify that \( v \approx 29.1375\, \text{m/s}\), consistent with the previous calculation.

Key Concepts

Wave Velocity

Wave velocity tells us how fast a wave travels through a medium. In the context of a wire, the velocity of a transverse wave can be calculated using the formula:\[ v = f \lambda \]where:

• \( v \) is the wave velocity in meters per second (m/s)
• \( f \) is the frequency of the wave in hertz (Hz)
• \( \lambda \) is the wavelength in meters (m)

Understanding wave velocity is crucial in applications like musical instruments, where the speed of waves on strings determines the pitch we hear. With a frequency of \( 875 \, \text{Hz} \) and a wavelength of \( 0.0333 \, \text{m} \), the wave velocity can be clearly computed as \( v = 29.1375 \, \text{m/s} \). This tells us how quickly disturbances propagate along the wire.

Tension in a Wire

Tension is a force stretching the wire linearly that's applied to either end. The tension in a wire affects the wave velocity, and can be calculated if you know the wave velocity and the linear mass density:\[ T = v^2 \mu \]Here:

• \( T \) is the tension in newtons (N)
• \( v \) is the wave velocity
• \( \mu \) is the linear mass density

In this exercise, with a wave velocity of \( 29.1375 \, \text{m/s} \) and linear mass density \( 0.022 \, \text{kg/m} \), the tension comes out to be around \( 18.67 \, \text{N} \). Ensuring correct tension is important for performance in structures and instruments made with wires or strings.

Linear Mass Density

Linear mass density \( \mu \) describes how much mass is distributed along a unit length of the wire:\[ \mu = \frac{m}{L} \]Where:

• \( m \) is the mass of the wire in kilograms (kg)
• \( L \) is the length of the wire in meters (m)

For the given wire with a mass of \( 0.0165 \, \text{kg} \) and a length of \( 0.75 \, \text{m} \), the linear mass density is computed as \( 0.022 \, \text{kg/m} \). Linear mass density plays a vital role in understanding how a wave travels through the wire — higher density wires will generally result in slower wave propagation.

Transverse Wave

Transverse waves are characterized by particle displacement perpendicular to the direction of wave travel. They're common in strings, where the vibration creates peaks and troughs as the wave propagates. Transverse waves are responsible for the musical notes produced by instruments like guitars and violins. In this exercise, we're looking at a transverse wave on a wire, with properties shaped by both tension and mass distribution:

• Unlike longitudinal waves, transverse waves can travel in media like strings and surfaces but not through fluids.
• The frequency and wavelength determine both the pitch and tone quality of notes played.
Understanding transverse waves helps us appreciate how vibrations translate into sound and how engineers can manipulate such properties to alter wave behaviors.