A vertical, 1.20 m length of 18 gauge (diameter of 1.024 mm ) copper wire has a 100.0 N ball

step 1 So they want us to find the wavelength of the third harmonic. This is very simple because the wa

A vertical, 1.20 m length of 18 gauge (diameter of 1.024...

A vertical, $1.20 \mathrm{~m}$ length of 18 gauge (diameter of $1.024 \mathrm{~mm}$ ) copper wire has a $100.0 \mathrm{~N}$ ball hanging from it. (a) What is the wavelength of the third harmonic for this wire? (b) A $500.0 \mathrm{~N}$ ball now replaces the original ball. What is the change in the wavelength of the third harmonic caused by replacing the light ball with the heavy one? (Hint: See Table 11.1 for Young's modulus.)

Key Concepts

Cross-Sectional Area of a Circular Wire

Many problems require calculating the cross-sectional area of a wire, typically assumed to have a circular cross-section. The area is given by A = ?(d/2)², where d is the diameter. This concept is critical when determining how the mechanical properties of the material, such as its resistance to deformation (as quantified by Young's modulus), influence the wire's extension under load.

Elastic Deformation and Young's Modulus

When a load is applied to a wire, it undergoes elastic deformation. Young's modulus (Y) is a material property that relates stress to strain and is used to calculate the elongation of the wire under tension. The extension can be determined from ?L = (T L?)/(A Y), where T is the tension, L? the original length, and A the cross-sectional area. This extension is important because any small change in the wire's length will affect the wavelengths of the standing waves it supports.

Wave Speed on a Tensioned String

The speed at which waves travel along a stretched string is determined by the tension in the string and its mass per unit length according to the formula v = ?(T/?). Although the wave speed does not affect the wavelength set by the fixed boundaries, it determines the frequencies of the standing waves and is crucial in understanding how changes in tension modify the dynamic behavior of the wire.

Standing Waves and Harmonics

Standing waves are formed on a string (or wire) fixed at both ends, and they exist only at specific frequencies and wavelengths that satisfy the boundary conditions. The wavelengths for these harmonics are given by the relation ?n = 2L/n, where L is the length of the string and n is the harmonic number. This concept is central when analyzing vibration modes for any tensioned string or wire, irrespective of its material.

Transcript

00:01 So they want us to find the wavelength of the third harmonic.

00:04 This is very simple because the wavelength of the third harmonic, or rather the wavelength of any harmonic is going to be equal to two times the length of the string time over the whatever harmonic it may be.

00:17 So over that n being an element of the set of all integers.

00:25 So n can only be an integer.

00:27 And we're trying to find the third harmonic.

00:30 So this will be 2l over 3, and this will be 2 times 1 .2 meters over 3.

00:41 So the wavelength of the third harmonic is simply going to be 0 .8 meters.

00:46 And then it's saying now there's another 100 -duton ball.

00:52 It's being replaced by 500 -newton ball.

00:55 This is what's creating the tension in the wire.

00:58 So if this ball is being replaced by a ball that has more mass, what's the change in wavelength of the third harmonic? so we can say that the young's modulus can be related to the force tension, whatever that force may be, times the length of the string, over the area, the cross -sectional area of the string, and then times the deformation or the change in length of the string.

01:34 So we can say that the change in length of the string is going to be equal to the force times the length of the string divided by the cross -sectional area times the young's modulus.

01:46 And we know that the area is going to be equal to pi r squared or pi d squared over.

01:59 Young's modulus is simply how it's also called a modulus of elasticity.

02:07 It's simply how resistant a string is or any material really, how resistant it is to deformation.

02:19 So f over a, that's a pressure, and then l over change in l, that's a strain.

02:25 So this is simply going to be basically how resistance it is to deformation, given that we're putting a much heavier ball at the end of the string.

02:39 So now we can say, okay, let's find, let's plug this cross -sectional area into this delta l...

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