Calculate Resonating Lengths for Vibrating String with Given Parameters

Math Problem Statement

Vibrating string Aim: Determine linear mass density of the wire, µ. Determination of the frequency of the Local A.C. mains. Determination of the velocity of the wave in Vibrating wire at the frequency of the Local A.C. mains

Apparatus Sonometer with wire, horseshoe magnet, small masses mains transformer Y [240V-6V], weighing balance.

Method Determine linear mass density of the wire, µ. To find the mass per meter, µ, of the wire you are given a cut known length X [at least 50cm] of the wire and weighs it carefully. If more convenient, µ can be found by measuring the diameter of the wire at three different places in perpendicular directions to the length of the wire and obtaining the density of the material from physical tables.

1 2 3 Mean

Length of wire X (m)

Mass, m, of X meters of wire, (kg)

Diameter of the wire (m)

Using appropriate formulae and method determine the, µ, of the wire.

Measurement of the frequency, f, of the Local A.C. mains Clean the wire X where it passes over the metal pulley P and at the far end where it is attached to sonometer S. Connect the wire, taking one lead from the pulley in series with the rheostat R and the 6V secondary of the mains transformer Y.Place the magnet G so that the wire passes between its poles and adjust the current [which may be checked with an ammeter] until the wire can be felt to vibrate slightly without becoming appreciably heated.Weigh the scale pan on the weighing balance and hang it on the wire. Adjust the positions of the bridges A and B until the wire between them resonates in its fundamental mode the magnet being at the center of AB. Measure the length L of the vibrating wire between A and B and record. Then Place a load of about 20g in the scale pan and get the combined mass M of the load and scale pan and repeat the above steps.Vary the load to about 120g in steps of 20g and each time measure the new resonating length.

Li near mass density of the wire,µ, =..g/m Mass Tension ,T. Length of fundamental mode ,L. L2

Scale pan

Draw a graph of T against L2 Using the graph determination of the frequency of the Local A.C. mains.

Determination of the velocity of the wave in the string: A.C. mains Now Hang a single mass of approximately 20g on the the wire Adjust the positions of the bridges A and B until the wire between them resonates in its fundamental mode ( 1 loop) the magnet being at the center of AB. Measure the length L of the vibrating wire between A and B and record. Then move the bridges A and B to get the second, third, fourth, fifth, and sixth .. loops. Each time measure the length L of the vibrating wire between A and B and record.

Tension on the wire=.N linear mass density of the wire,µ, =..g/m frequency of the wave=.. Hz.

No. of loops, N, Resonating length,L.

Draw a graph of L against N Using the graph determination of the velocity of the wave in the vibrating string at Local A.C. mains. IIgnore all the other questions and address this one:In determination of the velocity of the wave on the string:the tension on the wire=0.2N,μ =1.9431×10–4 Kg/m and frequency=30.17Hz .What will be the ideal values of the resonating length for 1,2,3,4,5,6,7 loops each

Solution

To calculate the resonating lengths $L$ for $N$ loops in a vibrating string, we use the relationship between the velocity of the wave $v$ , the frequency $f$ , and the wavelength $\lambda$ . The formula for the velocity is:

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

where:

$T$ is the tension in the wire (in newtons),
$\mu$ is the linear mass density of the wire (in kg/m).

Step 1: Calculate the velocity of the wave $v$

Substituting the given values:

$T = 0.2 \, \text{N}$ ,
$\mu = 1.9431 \times 10^{-4} \, \text{kg/m}$ ,

$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{0.2}{1.9431 \times 10^{-4}}}$

$v = \sqrt{1029.6} \approx 32.1 \, \text{m/s}.$

Step 2: Calculate the wavelength $\lambda$ for each mode

The wavelength for $N$ loops is related to the length $L$ and the number of loops $N$ by:

$\lambda = \frac{2L}{N}$

The relationship between velocity, frequency, and wavelength is:

$v = f \lambda \quad \Rightarrow \quad \lambda = \frac{v}{f}.$

Substituting $v = 32.1 \, \text{m/s}$ and $f = 30.17 \, \text{Hz}$ :

$\lambda = \frac{32.1}{30.17} \approx 1.064 \, \text{m}.$

Step 3: Calculate the resonating length $L$

For $N$ loops, the length $L$ is related to the wavelength $\lambda$ by:

$L = \frac{N \lambda}{2}.$

Substitute $\lambda = 1.064 \, \text{m}$ to calculate $L$ for each $N$ :

For $N = 1$ : $L = \frac{1 \times 1.064}{2} = 0.532 \, \text{m}.$
For $N = 2$ : $L = \frac{2 \times 1.064}{2} = 1.064 \, \text{m}.$
For $N = 3$ : $L = \frac{3 \times 1.064}{2} = 1.596 \, \text{m}.$
For $N = 4$ : $L = \frac{4 \times 1.064}{2} = 2.128 \, \text{m}.$
For $N = 5$ : $L = \frac{5 \times 1.064}{2} = 2.660 \, \text{m}.$
For $N = 6$ : $L = \frac{6 \times 1.064}{2} = 3.192 \, \text{m}.$
For $N = 7$ : $L = \frac{7 \times 1.064}{2} = 3.724 \, \text{m}.$

Final Results

The ideal resonating lengths for $N = 1, 2, 3, 4, 5, 6, 7$ loops are:

$L = 0.532 \, \text{m}, 1.064 \, \text{m}, 1.596 \, \text{m}, 2.128 \, \text{m}, 2.660 \, \text{m}, 3.192 \, \text{m}, 3.724 \, \text{m}.$

Would you like further clarifications or additional calculations?

Five relative questions:

How does the linear mass density $\mu$ affect the velocity of the wave?
Can you determine the frequency of the A.C. mains directly using the vibrating wire?
What is the relationship between tension and resonating frequency in a vibrating wire?
How does the number of loops influence the resonating length?
How would errors in measuring $\mu$ impact the calculated results?

Tip: Always ensure precise measurement of tension and mass to minimize errors in wave velocity calculations.

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Math Problem Analysis

Mathematical Concepts

Wave Mechanics
Harmonic Motion
Resonance

Formulas

Wave velocity: v = sqrt(T/μ)
Wavelength: λ = v/f
Resonating length: L = Nλ/2

Theorems

Wave Equation

Suitable Grade Level

Undergraduate Physics or Advanced High School (Grades 11-12)

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