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Chapter 10: Problem 32

The speed of a transverse wave, going on a wire having a length \(50 \mathrm{~cm}\) and mass \(5.0 \mathrm{~g}\), is \(80 \mathrm{~m} / \mathrm{s}\). The area of cross-section of the wire is \(1.0 \mathrm{~mm}^{2}\) and its Young's modulus is \(16 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\). Find the extension of the wire over its natural length.

Short Answer

Expert verified
The extension of the wire is 0.2 mm.

Step by step solution

01

Write Down Known Values

We are given the following values:1. Length of the wire, \( L = 50 \) cm = 0.5 m2. Mass of the wire, \( m = 5.0 \) g = 0.005 kg3. Speed of the wave, \( v = 80 \) m/s4. Cross-sectional area, \( A = 1.0 \text{ mm}^2 = 1.0 \times 10^{-6} \text{ m}^2 \)5. Young's modulus, \( Y = 16 \times 10^{11} \text{ N/m}^2 \)
02

Calculate Linear Mass Density

The linear mass density \( \mu \) is calculated using the formula \( \mu = \frac{m}{L} \).\[\mu = \frac{0.005 \text{ kg}}{0.5 \text{ m}} = 0.01 \text{ kg/m}\]
03

Relate Speed with Tension

The speed of a wave \( v \) is also related to tension \( T \) and linear mass density \( \mu \) by the formula \( v = \sqrt{\frac{T}{\mu}} \).\[T = \mu v^2 = 0.01 \times 80^2 = 64 \text{ N}\]
04

Use Hooke's Law for Extension

Using Hooke's Law, the relation between tension \( T \), extension \( \Delta L \), original length \( L \), area \( A \), and Young's modulus \( Y \) is given by \( T = \frac{Y \cdot A \cdot \Delta L}{L} \).Rearrange to find \( \Delta L \):\[\Delta L = \frac{T \cdot L}{Y \cdot A} = \frac{64 \cdot 0.5}{16 \times 10^{11} \cdot 1 \times 10^{-6}}\]\[\Delta L = 2 \times 10^{-4} \text{ m} = 0.2 \text{ mm}\]
05

Conclusion

The extension of the wire is \(0.2\) mm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transverse Waves
Transverse waves are a fundamental type of wave where the disturbance or oscillation occurs perpendicular to the direction of energy transfer.
Imagine a ripple in a pond or a wave on a string, where the movement goes up and down while the wave travels forward.
These waves are commonly seen in scenarios involving strings, rods, or any material capable of supporting side-to-side motion.
  • In the context of the exercise, transverse waves move along the wire.
  • The speed of these waves depends on the tension in the wire and its linear mass density.
Understanding transverse waves helps us analyze how energy is transmitted through materials without transporting the material itself over long distances.
Young's Modulus
Young's Modulus is an important concept that measures the stiffness of a material.
It is defined as the ratio of stress (force per unit area) to strain (proportional deformation) in a material.
This is expressed in the formula:\[ Y = \frac{stress}{strain} = \frac{F/A}{\Delta L/L} \]where:
  • \( F \) is the force applied,
  • \( A \) is the cross-sectional area,
  • \( \Delta L \) is the change in length,
  • \( L \) is the original length.
Young's Modulus helps us understand how materials will expand or compress under stress.
In the exercise, knowing the Young's Modulus of the wire allows us to calculate the extension when force is applied.
A higher Young's Modulus indicates that a material is harder to deform.
Hooke's Law
Hooke's Law is essential for understanding the elasticity of materials.
It states that, within the limit of proportionality, the extension of a material is directly proportional to the force applied to it.\[ F = k \cdot x \]where:
  • \( F \) is the force applied.
  • \( k \) is the spring constant (a measure of the stiffness).
  • \( x \) is the extension or compression length.
This law allows us to predict how much a wire will stretch under a given force, provided it doesn鈥檛 exceed its elastic limit.
Using the exercise, Hooke鈥檚 Law helps us relate the tension in the wire to the extension experienced due to the wave speed.
Linear Mass Density
Linear mass density is a measure of mass per unit length of an object.
It is especially useful when evaluating waves on a string or wire, such as transverse waves.\[ \mu = \frac{m}{L} \]where:
  • \( \mu \) is the linear mass density,
  • \( m \) is the mass of the wire,
  • \( L \) is the length of the wire.
In our exercise, we find the linear mass density to apply it in the wave speed formula.
This helps to determine the tension required for a certain wave speed.
It tells us how concentrated the mass is along the length of the wire, which in turn influences how easily waves can travel along it.

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Most popular questions from this chapter

Statement-1: When a source of sound is approaching an observer, the frequency perceived by the observer is greater than the frequency of the sound source. Statement-2: As the sound-source approaches the listener, the wavelength of the sound decreases while its speed relative to the medium remains unchanged.

(a) Lel at \(t=t^{\prime}, v=0\) \(\Rightarrow 0=100 \cos \left(20 t^{\prime}+\frac{\pi}{3}\right)\) \(\therefore\left(20 t^{\prime}+\frac{\pi}{3}\right)=\frac{\pi}{2}\) or \(\frac{3 \pi}{2}\) or \(\frac{5 \pi}{2}\) etc. for first come to rest, \(\left(20 t^{\prime}+\frac{\pi}{3}\right)=\frac{\pi}{2}\) \(\therefore \quad 20 t^{\prime}=\frac{\pi}{6} \quad \therefore \quad t^{\prime}=\frac{\pi}{120} \mathrm{~s}\) (b) Let at \(r^{\prime \prime}, a=0\) \(\therefore \quad 0=-2000 \sin \left(20 t^{\prime \prime}+\frac{\pi}{3}\right)\) \(\Rightarrow 20 r^{\prime \prime}+\frac{\pi}{3}=0\) or \(\pi\) or \(2 \pi\) or \(3 \pi \mathrm{ctc} .\) For first have zero acccleration, \(20 t^{\prime \prime}+\frac{\pi}{3}=\pi \quad \Rightarrow \quad 20 t^{\prime \prime}=\frac{2 \pi}{3}\) \(\therefore \quad t^{\prime \prime}=\frac{\pi}{30} \mathrm{~s}\) (c) For \(v^{\max }, \cos \left(20 t+\frac{\pi}{3}\right)=\pm 1\) \(\therefore 20 t+\frac{\pi}{3}=\pi\) or \(2 \pi\) or \(3 \pi\) etc. For first have maximum specd, \(20 t+\frac{\pi}{3}=\pi\) \(\Rightarrow \quad 20 t=\frac{2 \pi}{3} \quad \therefore \quad t=\frac{\pi}{30} \mathrm{~s}\)

A disc of radius \(20 \mathrm{~m}\) is rotating uniformly with angular frequency \(\omega=10 \mathrm{rad} / \mathrm{s}\), A source is fixod to rim of dise. The ratio of maximum and minimum froquency heard by observer far away from disc in plane of disc is (take speed of sound \(330 \mathrm{~m} / \mathrm{s}\) ) (a) \(33 / 13\) (b) \(33 / 53\) (c) \(13 / 53\) (d) \(53 / 13\)

Statement-1: For a particle executing SHM following equation can not be applied Displacement \(-\vec{v}_{w v} \times\) time Statement-2: SHM is not an example of uniformly accelerated motion.

With regard to sound waves, mark out corrcet suatements. (a) Phase difference between incident and reflected displacement waves from fixed end is zero. (b) Phase difference between incident and reflected pressure wave from fixed end is zero. (c) Phase difference between incident and reflected pressure wave from fixed end is \(\pi\). (d) Phase difference between incident and reflected displacement wave from fixed end is \(\pi\).
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