Problem 18 - Physics

Question

A steel cable of cross-sectional area \(2.83 \times 10^{-3} \mathrm{~m}^{2}\) is kept under a tension of \(1.00 \times 10^{4} \mathrm{~N}\). The density of steel is \(7860 \mathrm{~kg} / \mathrm{m}^{3}\) (this is not the linear density). At what speed does a transverse wave move along the cable?

Step-by-Step Solution

Answer

The speed of the transverse wave is approximately 21.2 m/s.

Step 1: Identify Relevant Formula

To find the speed of a transverse wave on a cable, we use the formula: \( v = \sqrt{\frac{T}{\mu}} \), where \( v \) is the speed of the wave, \( T \) is the tension in the cable, and \( \mu \) is the linear mass density of the cable.

Step 2: Calculate Linear Mass Density

The linear mass density \( \mu \) can be calculated using the formula \( \mu = \rho A \), where \( \rho \) is the density of steel and \( A \) is the cross-sectional area of the cable. Substitute given values: \( \mu = 7860 \, \text{kg/m}^3 \times 2.83 \times 10^{-3} \, \text{m}^2 = 22.24 \, \text{kg/m} \).

Step 3: Substitute Values into Wave Speed Formula

Substitute \( T = 1.00 \times 10^4 \, \text{N} \) and \( \mu = 22.24 \, \text{kg/m} \) into the wave speed formula: \( v = \sqrt{\frac{1.00 \times 10^4}{22.24}} \).

Step 4: Calculate the Wave Speed

Perform the calculation: \( v = \sqrt{\frac{1.00 \times 10^4}{22.24}} = \sqrt{449.28} \approx 21.2 \, \text{m/s} \).

Key Concepts

Wave Mechanics

Wave mechanics is a branch of physics that studies the behavior of waves. Here, we're particularly interested in transverse waves propagating through media, such as cables. Transverse waves are characterized by particle motion that is perpendicular to the direction of wave propagation. This differs from longitudinal waves, where particle motion is parallel to the wave's travel direction.
In the context of cables, understanding how a wave moves is vital for predicting vibrational characteristics. The speed at which a wave travels along the cable is influenced by properties like tension and mass distribution. Such an understanding is practical for engineering and materials science, allowing us to optimize structures and predict their responses to forces.
Calculating wave speed involves balancing the tension (force applied to keep the cable taut) and linear mass density (distribution of mass per unit length). When both are known accurately, the wave mechanics principles allow for effective determination of how fast a transverse wave can move along the medium.

Linear Mass Density

Linear mass density, denoted often by the symbol \( \mu \), is a measure of mass distribution along an object's length. It's a crucial factor for calculating wave speeds in mediums like cables.
The formula for linear mass density is \( \mu = \rho A \), where \( \rho \) is the material's density, and \( A \) is the cross-sectional area.

• High linear mass density signifies more mass per unit length; it usually results in slower wave speeds in a given material.
• Lower linear mass density implies less mass per unit length, facilitating faster wave speeds.

In the exercise, we calculated the linear mass density for a steel cable by multiplying its density (7860 kg/m³) with its cross-sectional area (2.83 × 10^-3 m²), yielding \( \mu = 22.24 \, \text{kg/m} \).
This value then becomes a vital component in calculating wave speed, helping characterize how external forces are distributed over the cable's length.

Tension in Cables

Tension refers to the force applied along the length of an object, such as a cable, which influences wave propagation. Tension is measured in newtons (N), and its magnitude directly affects the speed of waves moving through the cable.
The higher the tension, the greater the force pulling on the cable along its length, which can cause waves to travel faster. Conversely, if the tension is low, the wave speed decreases.

• In practical applications, like in bridges or elevators, maintaining optimal tension is crucial for stability and performance.
• For transverse wave speed calculations, tension is a key variable in the formula \( v = \sqrt{\frac{T}{\mu}} \).

In our exercise, the tension in the steel cable is given as \( 1.00 \times 10^4 \, \text{N} \), which is used to find the speed of the transverse wave moving through the cable. By understanding the tension, one can anticipate how fast the wave travels, ultimately aiding in designing resilient and secure infrastructure.