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Chapter 16: Problem 16

The speed of a transverse wave on a string is \(170 \mathrm{~m} / \mathrm{s}\) when the string tension is \(120 \mathrm{~N}\). To what value must the tension be changed to raise the wave speed to \(180 \mathrm{~m} / \mathrm{s} ?\)

Short Answer

Expert verified
The tension must be changed to approximately 134.37 N.

Step by step solution

01

Understand the Wave Speed Formula

The speed of a transverse wave on a string is given by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear mass density of the string. Our goal is to find the new tension \( T' \) when the wave speed is increased.
02

Write Equation for Initial Conditions

For the initial conditions, we have a wave speed \( v_1 = 170 \, \mathrm{m/s} \) and tension \( T_1 = 120 \, \mathrm{N} \). Plugging these values into the wave speed formula, we get \( 170 = \sqrt{\frac{120}{\mu}} \).
03

Solve for Linear Mass Density \( \mu \)

Rearrange your equation from Step 2 to solve for \( \mu \): \[\mu = \frac{120}{170^2} \]Solving, \( \mu = \frac{120}{28900} \approx 0.00415 \, \mathrm{kg/m} \).
04

Write Equation for New Conditions

For the new conditions, we want to find the tension \( T' \) when the wave speed \( v_2 = 180 \, \mathrm{m/s} \). Plugging the known values and \( \mu \) into the wave speed formula yields:\[180 = \sqrt{\frac{T'}{0.00415}} \]
05

Solve for New Tension \( T' \)

Rearrange the equation from Step 4 to solve for the new tension \( T' \): \[T' = 180^2 \times 0.00415\]Calculate \( T' \), which gives \( T' = 134.37 \, \mathrm{N} \).
06

Finalize the Solution

The new tension needed to raise the wave speed to \( 180 \, \mathrm{m/s} \) is approximately \( 134.37 \, \mathrm{N} \).

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Key Concepts

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

Transverse Wave
A transverse wave is a type of wave where the motion of the wave's medium is perpendicular to the direction of the wave's travel.
In other words, if the wave is moving horizontally, the particles in the medium move up and down.
This can be compared to how waves move on a string or a water surface, where the crest and trough move vertically while the wave itself moves horizontally. Some common examples of transverse waves include:
  • Light waves
  • Radio waves
  • Waves on strings
Understanding the nature of transverse waves is essential when dealing with physics problems involving wave speed, as it dictates the motion we're examining in strings and similar mediums.
String Tension
String tension is a crucial factor in determining the properties of waves on a string.
Tension refers to the force applied along the string, trying to elongate it.This tension influences the wave speed on the string, which is described by the formula:\[v = \sqrt{\frac{T}{\mu}}\]Where:
  • \( v \) is the wave speed
  • \( T \) is the tension in the string
  • \( \mu \) is the linear mass density
In the context of a physics problem, changing the tension will directly affect the speed of the transverse wave.
If you increase the tension, the wave speed will typically increase, as confirmed by the calculations in the provided solution. Understanding this principle helps solve many physics problems involving wave speed on strings.
Linear Mass Density
Linear mass density (\(\mu\)) is a key factor in analyzing waves on a string.
It is defined as the mass of the string per unit length, indicating how much mass is distributed along the length of the string. Mathematically, it is expressed as:\[\mu = \frac{m}{L}\]Where:
  • \( m \) is the mass of the string
  • \( L \) is the length of the string
In the context of the original exercise, knowing the linear mass density of the string allows you to relate tension changes to wave speed changes.
Once determined, it serves as a constant, helping to solve for new conditions with varying tensions and speeds.
Physics Problems
Physics problems involving waves often test your understanding of the relationship between wave properties, such as wave speed, tension, and mass density.
By carefully analyzing how each factor affects the wave's behavior, you'll be equipped to tackle similar exercises confidently. Some key strategies include:
  • Identifying known variables and constants.
  • Using the wave speed formula to relate tension and linear mass density with wave speed.
  • Ensuring units are consistent to prevent error during calculations.
In the given problem, these steps are demonstrated effectively to find the new string tension required to alter the wave speed.
Mastering such problems enhances your problem-solving skills and deepens your understanding of wave dynamics in diverse physics contexts.

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Most popular questions from this chapter

A sinusoidal wave is traveling on a string with speed \(40 \mathrm{~cm} / \mathrm{s}\). The displacement of the particles of the string at \(x=10 \mathrm{~cm}\) varies with time according to \(y=(5.0 \mathrm{~cm}) \sin \left[1.0-\left(4.0 \mathrm{~s}^{-1}\right) t\right] .\) The linear density of the string is \(4.0 \mathrm{~g} / \mathrm{cm}\). What are (a) the frequency and (b) the wavelength of the wave? If the wave equation is of the form \(y(x, t)=\) \(y_{m} \sin (k x \pm \omega t)\), what are (c) \(y_{m}\), (d) \(k\), (e) \(\omega\), and \((\mathrm{f})\) the correct choice of sign in front of \(\omega ?(\mathrm{~g})\) What is the tension in the string?

A sinusoidal wave travels along} a string. The time for a particular point to move from maximum displacement to zero is \(0.170 \mathrm{~s}\). What are the (a) period and (b) frequency? (c) The wavelength is \(1.40 \mathrm{~m}\); what is the wave speed?

A transverse sinusoidal wave is generated at one end of a long, horizontal string by a bar that moves up and down through a distance of \(1.00 \mathrm{~cm}\). The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 \(\mathrm{g} / \mathrm{m}\) and is kept under a tension of \(90.0 \mathrm{~N}\). Find the maximum value of (a) the transverse speed \(u\) and (b) the transverse component of the tension \(\tau\). (c) Show that the two maximum values calculated above occur at the same phase values for the wave. What is the transverse displacement \(y\) of the string at these phases? (d) What is the maximum rate of energy transfer along the string? (e) What is the transverse displacement \(y\) when this maximum transfer occurs? (f) What is the minimum rate of energy transfer along the string? (g) What is the transverse displacement \(y\) when this minimum transfer occurs?

93 A traveling wave on a string is described by $$ y=2.0 \sin \left[2 \pi\left(\frac{t}{0.40}+\frac{x}{80}\right)\right] $$ where \(x\) and \(y\) are in centimeters and \(t\) is in seconds. (a) For \(t=0\), plot \(y\) as a function of \(x\) for \(0 \leq x \leq 160 \mathrm{~cm} .\) (b) Repeat (a) for \(t=0.05 \mathrm{~s}\) and \(t=0.10 \mathrm{~s}\). From your graphs, determine (c) the wave speed and (d) the direction in which the wave is traveling.

A sinusoidal wave of angular frequency \(1200 \mathrm{rad} / \mathrm{s}\) and amplitude \(3.00 \mathrm{~mm}\) is sent along a cord with linear density \(2.00 \mathrm{~g} / \mathrm{m}\) and tension \(1200 \mathrm{~N}\). (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) \(0,(\mathrm{~d}) 0.4 \pi \mathrm{rad}\), and \((\mathrm{e}) \pi \mathrm{rad} ?\)
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