Problem 19 - University Physics with Modern Physics

Question

A thin, 75.0-cm wire has a mass of 16.5 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 cm makes 625 vibrations per second? (b) How fast would this wave travel?

Step-by-Step Solution

Answer

(a) The tension must be 9.52 N. (b) The wave travels at 20.81 m/s.

Step 1: Convert Units

First, convert all given measurements to standard SI units. The length of the wire is given as 75.0 cm, which is 0.750 meters. The mass of the wire is 16.5 g, which is 0.0165 kilograms. The wavelength is 3.33 cm, which is 0.0333 meters.

Step 2: Calculate Linear Mass Density

The linear mass density, \( \mu \), of the wire is the mass per unit length. Use the formula \( \mu = \frac{m}{L} \).Substitute the values:\( \mu = \frac{0.0165 \, \text{kg}}{0.750 \, \text{m}} = 0.022 \, \text{kg/m} \).

Step 3: Find Wave Speed Using Frequency and Wavelength

The wave speed, \( v \), can be determined using the formula \( v = f \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength.Given \( f = 625 \, \text{Hz} \) and \( \lambda = 0.0333 \, \text{m} \), the wave speed is:\( v = 625 \, \text{Hz} \times 0.0333 \, \text{m} = 20.81 \, \text{m/s} \).

Step 4: Calculate Tension in the Wire

Using the wave speed formula for a string, \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension, solve for \( T \): \( v^2 = \frac{T}{\mu} \).So, \( T = \mu v^2 \).Substitute \( \mu = 0.022 \, \text{kg/m} \) and \( v = 20.81 \, \text{m/s} \):\(T = 0.022 \, \text{kg/m} \times (20.81 \, \text{m/s})^2 = 9.52 \, \text{N} \).

Step 5: Calculate Wave Velocity Again (Validation)

Re-confirm the wave speed using the calculated tension to ensure calculations are correct.\( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{9.52 \, \text{N}}{0.022 \, \text{kg/m}}} \).Carrying out the calculation yields \( v = 20.81 \, \text{m/s} \), confirming initial calculation.

Key Concepts

Linear Mass Density

Linear mass density is a crucial concept in wave mechanics. It refers to how much mass is distributed along a unit length of a material, such as a wire or string. In this exercise, we calculate linear mass density using the formula: \[ \mu = \frac{m}{L} \] where \( m \) is the mass and \( L \) is the length of the wire. By substituting the given values, \( m = 0.0165 \) kg and \( L = 0.750 \) m, we find that \( \mu = 0.022 \) kg/m.
This measure tells us how the mass is spread out over the wire.
Understanding linear mass density helps in determining other properties like wave speed and tension.

Wave Speed

Wave speed is another fundamental aspect of wave mechanics. It indicates how fast a wave moves through a medium. The wave speed \( v \) is calculated using the relationship between frequency \( f \) and wavelength \( \lambda \): \[ v = f \lambda \] Given \( f = 625 \) Hz and \( \lambda = 0.0333 \) m, we find that \( v = 20.81 \) m/s.
This speed tells us how quickly the crests of the wave move across the wire.

• High frequency and long wavelength typically mean higher wave speed.
• Wave speed can affect sound and vibration transmission.

Tension in a Wire

Tension is the force exerted on a wire that influences wave propagation. To find the tension \( T \) in the wire that produces a specific wave speed, we use: \[ v = \sqrt{\frac{T}{\mu}} \]Rearrange to solve for tension:\[ T = \mu v^2 \]Given \( \mu = 0.022 \) kg/m and wave speed \( v = 20.81 \) m/s, we calculate \( T = 9.52 \) N.
This makes sure the wave travels accurately at the expected speed.

• Correct tension is vital for musical instruments and engineering structures.
• Altering tension can change wave speed and pitch of sound.

Transverse Wave

A transverse wave is one where the motion of the medium is perpendicular to the direction of the wave. When discussing waves on a wire, like in this exercise, we refer to transverse waves. Here, the wave travels along the wire while the wire moves up and down. Characteristics include:

• Crests and troughs as distinct features.
• Common in light waves and waves on strings.

Transverse waves are distinct from longitudinal waves, where the motion is parallel to the wave's direction. This concept helps visualize the wave's shape and behavior.