Problem 5 - College Physics

Question

A stecl wire 4.00 \(\mathrm{m}\) long has a mass of 0.0600 \(\mathrm{kg}\) and is stretched with a tension of 1000 \(\mathrm{N}\) . What is the speed of prop- agation of a transverse wave on the wire?

Step-by-Step Solution

Answer

The speed of the wave is approximately 258.2 m/s.

Step 1: Calculate Linear Density

To find the speed of the wave, we first calculate the linear density (mass per unit length) of the wire, represented by \( \mu \). This is calculated using the formula \( \mu = \frac{m}{L} \) where \( m = 0.0600 \, \text{kg} \) is the mass of the wire and \( L = 4.00 \, \text{m} \) is the wire's length. Plugging in the values, we get \( \mu = \frac{0.0600}{4.00} = 0.0150 \, \text{kg/m} \).

Step 2: Use the Wave Speed Formula

The speed of a transverse wave on a wire is given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T = 1000 \, \text{N} \) is the tension and \( \mu = 0.0150 \, \text{kg/m} \) is the linear density. Substitute these values into the equation to get \( v = \sqrt{\frac{1000}{0.0150}} \).

Step 3: Calculate the Square Root

Calculate the division first: \( \frac{1000}{0.0150} = 66666.67 \). Then, find the square root: \( v = \sqrt{66666.67} \approx 258.2 \, \text{m/s} \).

Key Concepts

Linear Density

In the context of waves on a wire, understanding linear density is essential. Linear density, represented as \( \mu \), is the mass of the wire per unit length. It provides a means to relate mass and length in wave calculations.
To calculate linear density, use the formula \( \mu = \frac{m}{L} \). Here, \( m \) is the mass of the wire and \( L \) is its length.
In the exercise example, the mass \( m = 0.0600 \, \text{kg} \) and the length \( L = 4.00 \, \text{m} \). Thus, the linear density is \( \mu = \frac{0.0600}{4.00} = 0.0150 \, \text{kg/m} \). This value is crucial for further steps in calculating wave speed.

Wave Speed Formula

To find the speed of a transverse wave on a wire, we use the wave speed formula \( v = \sqrt{\frac{T}{\mu}} \). This formula highlights the relationship between tension, linear density, and wave speed.

• \( v \) is the wave speed
• \( T \) is the tension in the wire
• \( \mu \) is the linear density, calculated earlier

For the given problem, the tension \( T = 1000 \, \text{N} \) and the linear density \( \mu = 0.0150 \, \text{kg/m} \).
By substituting these values, the equation becomes \( v = \sqrt{\frac{1000}{0.0150}} \). The solution to this equation provides the wave speed.

Tension in a Wire

Tension is a key factor in wave propagation on a wire. In this exercise, the tension is given as \( 1000 \, \text{N} \).
Tension affects how quickly a wave travels along the wire. Higher tension results in faster wave speed, while lower tension slows it down.
It is crucial for determining wave speed through the wave speed formula \( v = \sqrt{\frac{T}{\mu}} \). Thus, the accuracy of tension measurements directly impacts the calculation of wave speed.

Mass Per Unit Length

Mass per unit length is synonymous with linear density, \( \mu \). It helps in understanding how much mass is distributed along each meter of the wire. The calculation itself involves dividing the wire's mass by its length.
Knowing the mass per unit length is important in physics, especially in wave mechanics. It informs the behavior of waves on strings, wires, and even in air columns.
In scenarios like the exercise example, calculating mass per unit length provides the necessary data for further determining how fast a wave can travel across a medium such as a tensioned wire.