Problem 18 - College 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 of 0.300 cm at the antinodes. (a)What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire.

Step-by-Step Solution

Answer

(a) The speed is 96.0 m/s. (b) The tension is 460.8 N.

Step 1: Understanding the Fundamental Frequency

The fundamental frequency, also known as the first harmonic, is the lowest frequency at which a system vibrates. For a string fixed at both ends, the length of the string is equal to half the wavelength of the fundamental mode. Given that the frequency is 60.0 Hz and the length of the wire is 80.0 cm or 0.8 m, we can derive further calculations.

Step 2: Calculate Wavelength for Fundamental Mode

Use the formula for wavelength \( \lambda \) in fundamental mode: \( \lambda = 2L \). Substituting the length \( L = 0.8 \) m, we get: \( \lambda = 2 \times 0.8 = 1.6 \) m.

Step 3: Calculate Speed of Propagation of Waves

The speed \( v \) of wave propagation on a string is given by the formula \( v = f \times \lambda \), where \( f \) is the frequency. Using the given frequency \( f = 60.0 \) Hz and the calculated wavelength \( \lambda = 1.6 \) m, we find: \[ v = 60.0 \times 1.6 = 96.0 \, \text{m/s}. \]

Step 4: Relate Wave Speed to Tension

The speed of a wave on a string is related to the tension \( T \) and linear mass density \( \mu \) by the formula \( v = \sqrt{\frac{T}{\mu}} \). Here, \( \mu = \frac{m}{L} \) is the mass per unit length of the wire.

Step 5: Calculate Linear Mass Density

The mass of the wire is 40.0 g, which is 0.040 kg. The length of the wire is 0.8 m. Therefore, the linear mass density \( \mu \) is \( \mu = \frac{0.040}{0.8} = 0.050 \, \text{kg/m}. \)

Step 6: Solve for Tension in the Wire

From the wave speed equation \( v = \sqrt{\frac{T}{\mu}} \), solve for tension \( T \): \[ T = v^2 \times \mu. \] Using \( v = 96.0 \) m/s and \( \mu = 0.050 \) kg/m, we get \[ T = (96.0)^2 \times 0.050 = 460.8 \, \text{N}. \]

Key Concepts

Understanding Fundamental Frequency

The **fundamental frequency** of a wave is the simplest and most basic pattern of vibration for a system. Think of it as the lowest hum a string can produce when plucked. For a wire fixed at both ends, this frequency corresponds to the situation where the string vibrates in one single "loop." Nothing more complex happens, just one easy-going wave traveling back and forth.
To understand its calculation, you should know that the characteristic wavelength for this frequency, given both ends are fixed, is twice the length of the wire. That's because the wire must contain exactly half of a full wave, considering it looks like a big arch between its endpoints.
In the problem we have, the frequency is given as 60.0 Hz and the wire length as 0.8 meters. Applying the formula for wavelength—\( \lambda = 2L \)—we find that the wavelength is 1.6 meters. From here, you can calculate other properties like wave speed.

Comprehending Tension in a Wire

**Tension in a wire** refers to the force that is pulling the wire taut. It is vital because this force determines how quickly waves can travel along the wire. The basic idea is the tighter the wire, the faster the waves will move.
How do we calculate this tension? We use the relationship between wave speed, tension, and linear mass density. This relationship is given by the formula:

• \( v = \sqrt{\frac{T}{\mu}} \)

where \( v \) is the speed of wave propagation, \( T \) is tension, and \( \mu \) is the linear mass density of the wire.
From this equation, rearranging gives us a way to determine tension if we know the wave speed and mass density.

• \( T = v^2 \times \mu \)

By applying given and calculated values in the problem, using a wave speed of 96 m/s and a mass density of 0.050 kg/m, we find the tension to be approximately 460.8 newtons.

Explaining Linear Mass Density

**Linear mass density** is a way to describe how much mass is distributed along a certain length of wire or string. If you imagine cutting a string into tiny pieces, each piece's mass would be part of its mass density. It's measured in kilograms per meter (kg/m).
For calculating it, divide the total mass of the wire by its length. In our scenario, the wire's mass is 40.0 g (or 0.040 kg). The full length we're dealing with is 0.8 meters.
Using the formula:

• \( \mu = \frac{m}{L} \)

we get a linear mass density of 0.050 kg/m. This value is essential when you need to link the mass properties of the string to the dynamic characteristics of waves traveling through it. Knowing \( \mu \) helps in determining how variables like tension affect wave speed.