What is the Speed of Transverse Wave on a Stretched String?

S7-SA4-0291

Grade Level:

Class 12

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Definition

What is it?

The speed of a transverse wave on a stretched string tells us how fast a disturbance, like a ripple, travels along the string. It depends on how much the string is stretched (tension) and how heavy it is per unit length (linear mass density).

Simple Example

Quick Example

Imagine you are playing a guitar. When you pluck a string, a wave travels along it. The speed at which this wave moves depends on how tightly you've tuned the string (tension) and if it's a thin string or a thick string (linear mass density). A tighter string makes the wave travel faster, and a lighter string also makes it travel faster.

Worked Example

Step-by-Step

Let's find the speed of a transverse wave on a guitar string.

Step 1: Identify the given values. Suppose the tension (T) in the string is 100 Newtons (N) and its linear mass density (mu) is 0.01 kg/m.

Step 2: Recall the formula for the speed of a transverse wave on a stretched string: v = sqrt(T / mu).

Step 3: Substitute the values into the formula: v = sqrt(100 N / 0.01 kg/m).

Step 4: Calculate the division inside the square root: 100 / 0.01 = 10000.

Step 5: Calculate the square root: v = sqrt(10000).

Step 6: The speed (v) is 100 m/s.

Answer: The speed of the transverse wave on the string is 100 meters per second.

Why It Matters

Understanding wave speed is crucial in many fields. In engineering, it helps design musical instruments and suspension bridges to prevent unwanted vibrations. It's also vital in space technology for understanding satellite communication and in medicine for ultrasound imaging, showing how waves move through different body tissues.

Common Mistakes

MISTAKE: Confusing linear mass density with total mass. Students sometimes use the total mass of the string in the formula. | CORRECTION: Linear mass density (mu) is mass per unit length (kg/m), not the total mass. Always divide the total mass by the string's length to get mu.

MISTAKE: Using tension in grams or kilograms. Tension is a force. | CORRECTION: Tension (T) must always be in Newtons (N). If given in grams or kilograms, convert it to Newtons by multiplying by 'g' (acceleration due to gravity, approx 9.8 m/s^2).

MISTAKE: Forgetting the square root. Students sometimes calculate T/mu and present that as the speed. | CORRECTION: The formula is v = sqrt(T/mu). Always remember to take the square root of the ratio of tension to linear mass density.

Practice Questions

Try It Yourself

QUESTION: A stretched wire has a tension of 49 N and a linear mass density of 0.01 kg/m. What is the speed of the transverse wave on the wire? | ANSWER: 70 m/s

QUESTION: If a string has a total mass of 0.02 kg and a length of 2 meters, and is under a tension of 80 N, what is the speed of the transverse wave? | ANSWER: 89.44 m/s

QUESTION: A string vibrates with a transverse wave speed of 50 m/s. If the linear mass density is 0.005 kg/m, what is the tension in the string? | ANSWER: 12.5 N

MCQ

Quick Quiz

Which of the following will increase the speed of a transverse wave on a stretched string?

Decreasing the tension in the string

Increasing the linear mass density of the string

Increasing the tension in the string

Making the string longer while keeping its total mass constant

The Correct Answer Is:

The formula for wave speed is v = sqrt(T/mu). Increasing tension (T) will increase the numerator, thus increasing the speed. Increasing linear mass density (mu) or decreasing tension would decrease the speed.

Real World Connection

In the Real World

This concept is used by musicians and instrument makers in India, like those who make sitars or tablas. They adjust the tension of the strings or drumheads to get the correct pitch, which means changing the speed of the waves produced. Also, in earthquake engineering, understanding how waves travel through structures helps design buildings in cities like Delhi and Mumbai to withstand tremors.

Key Vocabulary

Key Terms

TENSION: The pulling force acting along the string, measured in Newtons (N). | LINEAR MASS DENSITY: The mass of the string per unit of its length, measured in kg/m. | TRANSVERSE WAVE: A wave in which the particles of the medium oscillate perpendicular to the direction of wave propagation. | PROPAGATION: The process of transmitting or spreading something, like a wave, through a medium.

What's Next

What to Learn Next

Now that you understand wave speed, you can explore how these waves create sounds by learning about 'Standing Waves on a String'. This will help you understand how musical instruments produce different notes and how wave properties like frequency and wavelength are related.