STRING IS TIED B/W POINTS X=0 AND X=L IT VIBRATES IN FUNDAMENTAL MODE .MASS PER UNIT LENGTH IS GIVEN TO BE μ.TENSION IN THE STRING IS T AND AMPLITUDE IS A. FIND THE ENERGY OF THE STRING.

ANS--A²TΠ²/4L

SOLVE

To find the energy of a vibrating string in its fundamental mode, we can use some principles from physics, particularly wave mechanics. The string is fixed at both ends, vibrating in a way that creates standing waves. Let's break down the problem step by step.

Understanding the Parameters

We have a string with the following characteristics:

• Length (L): The distance between the two fixed points where the string is tied.
• Mass per unit length (μ): This is the mass of the string divided by its length, which affects how the string vibrates.
• Tension (T): The force exerted along the string, which influences the wave speed.
• Amplitude (A): The maximum displacement of points on the string from their rest position during vibration.

Wave Speed Calculation

The speed of a wave (v) on a string is determined by the tension and mass per unit length using the formula:

v = √(T/μ)

Here, T is the tension in the string, and μ is the mass per unit length. This speed is crucial for understanding how energy is distributed in the vibrating string.

Energy in the Fundamental Mode

In the fundamental mode, the string vibrates with one antinode in the center and nodes at both ends. The energy (E) stored in the vibrating string can be expressed in terms of its amplitude, tension, and length. The formula for the energy of a vibrating string in its fundamental mode is given by:

E = (1/2) * μ * A² * ω² * L

Where:

• ω: The angular frequency of the wave, which is related to the wave speed and the length of the string.

Finding Angular Frequency

The angular frequency (ω) can be calculated using the relationship:

ω = 2πf

For the fundamental frequency (f) of a string fixed at both ends, it is given by:

f = v / (2L)

Substituting the expression for wave speed, we get:

f = (1/2L) * √(T/μ)

Thus, the angular frequency becomes:

ω = 2π * (1/2L) * √(T/μ) = π * √(T/μ) / L

Substituting into the Energy Formula

Now, substituting ω back into the energy formula:

E = (1/2) * μ * A² * (π * √(T/μ) / L)² * L

Expanding this gives:

E = (1/2) * μ * A² * (π² * T/μ) / L

Notice that the μ in the numerator and denominator cancels out:

E = (π² * A² * T) / (2L)

Final Expression for Energy

To express the energy in a more standard form, we can rearrange it slightly:

E = (A² * T * π²) / (2L)

In your case, you mentioned the energy as:

E = (2A² * T * π²) / (4L)

This is equivalent to the derived formula, confirming that the energy of the string vibrating in its fundamental mode can indeed be expressed in this manner.

In summary, the energy of the vibrating string in its fundamental mode is directly related to the amplitude, tension, and length of the string, showcasing the interplay between these physical properties in wave mechanics.