Question

Verify that the quantity \(c=\sqrt{T / \mu}\) that appears in the wave equation for a string does indeed have the units of a speed.

Short answer

The quantity \(c=\sqrt{T / \mu}\) has the units of speed (m/s).

Step-by-step solution

Identify Units

Identify the units for the tension \(T\) and linear density \(\mu\) from the given expression \(c=\sqrt{T / \mu}\). Tension \(T\) is in units of force (Newtons), and linear density \(\mu\) is in units of mass per length (kg/m).

Break Down Units

Express the units of each component. The unit of \(T\) is \(N = \text{kg}\cdot\text{m/s}^2\) and \(\mu\) is \(\text{kg/m}\).

Calculate Units of \(T/\mu\)

Divide the units of \(T\) by the units of \(\mu\): \[\frac{\text{N}}{\text{kg/m}} = \frac{\text{kg}\cdot\text{m/s}^2}{\text{kg}/\text{m}} = \text{m}^2/\text{s}^2\]

Apply Square Root

Apply the square root to the units obtained in the previous step: \[\sqrt{\text{m}^2/\text{s}^2} = \text{m/s}\]. This reveals the quantity has units of speed.

Key concepts

Tension in a String

When we discuss tension in the context of physics and specifically strings, we're talking about a force. This force is applied uniformly throughout the entire length of the string, and it attempts to stretch or pull the string apart.
Tension is a critical factor in understanding wave motion in a string, as it affects how quickly waves can travel. The greater the tension, the faster the wave propagates along the string.

• Tension is measured in Newtons (N).
• It plays into the formula for wave speed \[c = \sqrt{\frac{T}{\mu}}\].
• High tension means faster wave speed.

> Imagine holding the ends of a rope and pulling tightly. The force you exert pulls the rope taut, and this is the tension acting within the rope.

This tension determines the energy with which waves can travel. Keep in mind that altering the tension changes the wave behavior significantly.

Understanding Linear Density

Linear density is a concept that comes into play when we talk about waves on a string. It is defined as the mass per unit length of the string or any one-dimensional object.
This measurement helps to understand how mass is distributed along the length of the string, which directly impacts the wave speed.

• Linear density is represented as \(\mu\) and its units are \(\text{kg/m}\).
• It explains how concentrated the string's mass is along its length.
• lterations in linear density change how waves behave on the string \(c = \sqrt{\frac{T}{\mu}}\).

> Just like how different thicknesses of a rope have different masses, high linear density means more mass per length, which affects wave travel speeds.

Hence, in the formula for calculating wave speed, linear density plays a crucial role in relation to tension.

Exploring the Units of Speed

Units of speed are fundamental to fully grasp concepts like wave propagation in physics. Speed is defined as the distance covered over a certain period. For wave speed along a string, the unit is typically meters per second (m/s).
Through the given wave equation for a string where speed \(c\) is calculated as \( c = \sqrt{\frac{T}{\mu}} \), understanding the units of tension and linear density helps see why the result is m/s.

• The unit of tension \(T\) is \( \text{kg}\cdot\text{m/s}^2\).
• Linear density \(\mu\) is expressed as \(\text{kg/m}\).
• When combined in the wave equation formula: \[ \frac{\text{N}}{\text{kg/m}} = \frac{\text{kg}\cdot\text{m/s}^2}{\text{kg/m}} = \text{m}^2/\text{s}^2 \]
• The square root turns \[ \text{m}^2/\text{s}^2 \] into \[ \text{m/s} \].

> This result, m/s, expresses speed, conclusively showing that the division of tension by linear density within the square root yields a speed measure.

Understanding this unit conversion is key in using the wave equation effectively across various physical scenarios.