Chapter 16: Problem 12
What must be the stress \((F/A)\) in a stretched wire of a material whose Young's modulus is \(Y\) for the speed of longitudinal waves to equal 30 times the speed of transverse waves?
Short Answer
Expert verified
The stress must be \( \frac{Y}{900} \).
Step by step solution
01
Understanding Wave Speeds
The speed of longitudinal waves in a material is given by \( v_l = \sqrt{\frac{Y}{\rho}} \), where \( Y \) is Young's modulus and \( \rho \) is the density of the material. The speed of transverse waves \( v_t \) in a wire is given by \( v_t = \sqrt{\frac{F}{\mu}} \), where \( F \) is the tension in the wire and \( \mu \) (linear density) is \( \rho \cdot A \) for a wire of cross-sectional area \( A \).
02
Expressing the Given Condition
According to the problem, the speed of longitudinal waves is 30 times the speed of transverse waves, hence \( v_l = 30 \cdot v_t \). We can substitute the expressions for \( v_l \) and \( v_t \), leading to: \[ \sqrt{\frac{Y}{\rho}} = 30 \cdot \sqrt{\frac{F}{\mu}} \] .
03
Solving for the Tension-to-Area Ratio
Equate the squares of both sides of the equation obtained: \( \frac{Y}{\rho} = 30^2 \cdot \frac{F}{\rho \cdot A} \). Note that \( \mu = \rho \cdot A \).
04
Cancelling and Rearranging Terms
Simplify the equation: \( \frac{Y}{\rho} = 900 \cdot \frac{F}{\rho \cdot A} \). Cancel \( \rho \) from both sides to get \( Y = 900 \cdot \frac{F}{A} \).
05
Finding the Stress
Rearrange the equation to solve for the stress \( \frac{F}{A} \): \( \frac{F}{A} = \frac{Y}{900} \).
06
Conclusion
The stress required in the wire for the condition given in the problem is \( \frac{Y}{900} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Speed
Wave speed is an essential concept in physics that tells us how quickly a wave travels through a medium. In the context of this exercise, we are dealing with two types of wave speeds: longitudinal and transverse.
For longitudinal waves in a material, the speed, denoted as \( v_l \), is derived from the equation:
On the other hand, the speed of transverse waves, \( v_t \), in a wire is expressed as:
Understanding these wave speeds is crucial for solving problems related to wave motion.
For longitudinal waves in a material, the speed, denoted as \( v_l \), is derived from the equation:
- \( v_l = \sqrt{\frac{Y}{\rho}} \)
On the other hand, the speed of transverse waves, \( v_t \), in a wire is expressed as:
- \( v_t = \sqrt{\frac{F}{\mu}} \)
Understanding these wave speeds is crucial for solving problems related to wave motion.
Stress in a Wire
Stress in a wire is an important concept when dealing with the elasticity of materials. Stress is the force \( F \) applied per unit area \( A \), represented by the equation:
To solve this, we use the equations for wave speeds and set up the equation:
- Stress \( = \frac{F}{A} \)
To solve this, we use the equations for wave speeds and set up the equation:
- \( \sqrt{\frac{Y}{\rho}} = 30 \cdot \sqrt{\frac{F}{\mu}} \)
- Stress \( = \frac{Y}{900} \)
Longitudinal Waves
Longitudinal waves are waves where the particles in the material move in the same direction as the wave. In the context of a wire, these waves occur when a material is compressed and stretched along the axis of the wave.
The speed at which these waves move through a medium depends on the material's properties, specifically its Young's modulus \( Y \) and density \( \rho \). The formula to compute this speed is:
The speed at which these waves move through a medium depends on the material's properties, specifically its Young's modulus \( Y \) and density \( \rho \). The formula to compute this speed is:
- \( v_l = \sqrt{\frac{Y}{\rho}} \)
Transverse Waves
Transverse waves, unlike longitudinal waves, involve particle motion perpendicular to the direction of wave propagation. In a wire, such waves occur when it undergoes displacement sideways under tension.
The speed of transverse waves \( v_t \) is determined by the tension in the wire \( F \) and its linear density \( \mu \). The expression for the speed is:
These waves are slower than longitudinal waves in the same material due to the nature of their particle motion. This difference in speed helps us understand why the stress in the wire needs to be precisely calculated to achieve the exercise's requirements.
The speed of transverse waves \( v_t \) is determined by the tension in the wire \( F \) and its linear density \( \mu \). The expression for the speed is:
- \( v_t = \sqrt{\frac{F}{\mu}} \)
These waves are slower than longitudinal waves in the same material due to the nature of their particle motion. This difference in speed helps us understand why the stress in the wire needs to be precisely calculated to achieve the exercise's requirements.