Speed of a Transverse Wave: Comprehensive NEET Physics Notes

1. Speed of a Transverse Wave

1.1 Introduction to Transverse Waves

Transverse waves are mechanical waves where the oscillation of particles in the medium occurs perpendicular to the direction of wave propagation. Examples include waves on a stretched string, electromagnetic waves, and water waves. Understanding the speed at which these waves travel is crucial for NEET aspirants, as it is a fundamental concept in wave mechanics.

Key Concept: The speed of a transverse wave depends on two primary factors:

1. The tension of the medium.
2. The linear mass density of the medium.

1.2 Formula for Speed of a Transverse Wave

The speed of a transverse wave on a stretched string is given by:

$v=\sqrt{\frac{T}{\mu }}$

where:

• $v$ = speed of the wave (m/s)
• $T$ = tension in the string (N)
• $\mu$ = linear mass density (kg/m), which is the mass per unit length of the string.

This formula shows that as the tension increases, the speed of the wave increases, and as the linear mass density increases, the speed decreases.

NEET Tip:

Memorize this formula: $v=\sqrt{\frac{T}{\mu }}$, as it frequently appears in NEET questions related to wave speed.

1.3 Derivation of the Speed Formula

The derivation of the formula involves using Newton's second law and understanding that the speed of a wave depends on the restoring force and inertial properties of the medium.

1. The restoring force is provided by the tension, $T$, in the string.
2. The inertial property is given by the linear mass density, $\mu$.

Using dimensional analysis, one can conclude that:

$v\propto \sqrt{\frac{T}{\mu }}$

Thus, introducing the dimensionless constant of proportionality, we arrive at the formula:

$v=\sqrt{\frac{T}{\mu }}$

Did You Know?

The speed of a transverse wave is independent of the wave’s frequency or wavelength. It depends solely on the medium's properties.

1.4 Factors Affecting the Speed of a Transverse Wave

1. Tension (T): As tension increases, the wave speed increases because the restoring force is stronger.
2. Linear Mass Density (μ): As linear mass density increases, the wave speed decreases due to increased inertia.

1.5 Real-Life Application

In musical instruments like guitars and violins, tightening the strings increases the tension, which increases the speed of transverse waves, resulting in a higher pitch.

1.6 Visual Illustration

(Diagram to be included): A labeled diagram showing a stretched string with a transverse wave traveling along it. Indicate the direction of the wave, oscillation direction of particles, and the parameters of tension (T) and mass density (μ) to visualize the speed formula.

1.7 Example Calculation

Problem: A string of length 0.72 m has a mass of 5.0 × 10⁻³ kg and is under a tension of 60 N. Calculate the speed of the transverse wave on this string.

Solution:

1. Calculate linear mass density: $\mu =\frac{\text{mass}}{\text{length}}=\frac{5.0\times 10^{-3}}{0.72}=6.9\times 10^{-3}\text{kg/m}$
2. Use the formula: $v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{60}{6.9\times 10^{-3}}}\approx 93\text{m/s}$

NEET Problem-Solving Strategy:

Always identify and calculate the tension and linear mass density first when solving speed problems.

Quick Recap

• The speed of a transverse wave is given by: $v=\sqrt{\frac{T}{\mu }}$
• As tension increases, the wave speed increases.
• As linear mass density increases, the wave speed decreases.

Practice Questions

Question 1

A wire of length 1.5 m and mass 0.015 kg is under a tension of 75 N. Calculate the speed of a transverse wave on this wire.

Solution:

1. Find linear mass density: $\mu =\frac{0.015}{1.5}=0.01\text{kg/m}$
2. Calculate speed: $v=\sqrt{\frac{75}{0.01}}=\sqrt{7500}\approx 86.6\text{m/s}$

Question 2

If the tension in a string is quadrupled, how does the speed of a transverse wave change?

Answer: The speed will double since $v\propto \sqrt{T}$.

Question 3

A 2 m long string has a mass of 0.02 kg and is under a tension of 20 N. What is the speed of the transverse wave?

Solution:

1. $\mu =\frac{0.02}{2}=0.01\text{kg/m}$
2. $v=\sqrt{\frac{20}{0.01}}=\sqrt{2000}\approx 44.7\text{m/s}$

Question 4

If the mass of a string is doubled but the tension remains constant, how is the speed affected?

Answer: The speed is reduced by a factor of $\sqrt{2}$.

Question 5

A string with a tension of 10 N has a transverse wave traveling at 50 m/s. What is the linear mass density of the string?

Solution: $\mu =\frac{T}{v^2}=\frac{10}{50^2}=0.004\text{kg/m}$

Concept Connection

Link to Biology: The concept of wave speed is similar to the propagation of action potentials in neurons, where the speed of nerve impulses depends on factors like axon diameter and myelination.

Link to Chemistry: Understanding wave speed is essential in studying the vibrations of molecules and how they absorb or emit energy in spectroscopy.

NEET Exam Strategy

• Practice problems involving varying tension and mass density.
• Understand the derivation and dependencies of wave speed on medium properties.
• Pay attention to units and conversions in calculations.

Glossary

• Transverse Wave: A wave in which particles of the medium move perpendicular to the direction of propagation.
• Linear Mass Density (μ): The mass per unit length of a string or medium.
• Tension (T): The force applied along the string, causing it to stretch.