Problem 86 - University Physics with Modern Physics

Question

A metal wire, with density \(\rho\) and Young's modulus \(Y\) is stretched between rigid supports. At temperature \(T,\) the speed of a transverse wave is found to be \(v_{1}\) . When the temperature is increased to \(T+\Delta T,\) the speed decreases to \(v_{2} < v_{1} .\) Determine the coefficient of linear expansion of the wire.

Step-by-Step Solution

Answer

\( \alpha = \frac{1}{\Delta T} \left( 1 - \frac{v_2^2}{v_1^2} \right) \)

Step 1: Understand the Relationship Between Wave Speed and Tension

The speed of a transverse wave on a wire is given by \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension, and \( \mu \) is the linear mass density \( \mu = \rho A \) (with \( A \) being the cross-sectional area). Since the ends are fixed, a thermal expansion occurs which changes the tension in the wire.

Step 2: Express Tension in Terms of Modulus of Elasticity

The tension in the wire, which affects the wave speed, can be expressed using Young's Modulus \( Y \). This is given by \( T = Y \times A \times \frac{\Delta L}{L} \), where \( \Delta L \) is the elongation due to thermal expansion.

Step 3: Define the Relationship of Wave Speed with Temperature Change

When the temperature increases, the length change due to the coefficient of linear expansion \( \alpha \) is \( \Delta L = \alpha L \Delta T \). Substitute into the expression for tension: \( T = Y \times A \times \alpha \Delta T \). Consequently, as the temperature increases the speed of the wave changes as \( v_2 = \sqrt{\frac{T_2}{\mu}} \), and originally \( v_1 = \sqrt{\frac{T_1}{\mu}} \).

Step 4: Solve for the Coefficient of Linear Expansion

Using the wave speeds, we know \( v_1^2 = \frac{Y A}{\rho A} \) initially and modifies to \( v_2^2 = \frac{Y A (1 - \alpha \Delta T)}{\rho A} \). Simplifying gives \( \alpha \Delta T = 1 - \frac{v_2^2}{v_1^2} \), thus \( \alpha = \frac{1}{\Delta T}\left(1 - \frac{v_2^2}{v_1^2}\right) \).

Step 5: Final Expression and Verification

So, the coefficient of linear expansion is calculated as \( \alpha = \frac{1 - \frac{v_2^2}{v_1^2}}{\Delta T} \). This equation is derived based on how linear expansion impacts tension and results in a change in wave speed.

Key Concepts

Wave Speed

Understanding wave speed is essential when studying physics and materials science. Wave speed, or the velocity of a wave, is calculated by dividing the distance that the wave travels by the time it takes to travel that distance. For a transverse wave on a wire, the formula is given by:

• \( v = \sqrt{\frac{T}{\mu}} \)

where \( v \) is the wave speed, \( T \) is the tension in the wire, and \( \mu \) is the linear mass density.
The tension in the wire pulls it taut, while the linear mass density relates to the mass per unit length of the wire.
Wave speed can change due to factors like tension adjustments and thermal expansion, as observed when temperature varies. This is because a rise in temperature causes the wire to expand, thereby altering the tension that directly affects the wave speed.
Thus, observing the change in wave speed at different temperatures helps to deduce the coefficient of linear expansion.

Young's Modulus

Young's modulus is a fundamental concept in elasticity, providing insight into a material's ability to resist changes in length when under tension or compression. It quantifies a material's stiffness or rigidity, calculated as:

• \( Y = \frac{T}{A \cdot \frac{\Delta L}{L}} \)

where \( Y \) is Young's modulus, \( T \) is the tension force, \( A \) is the cross-sectional area, \( \Delta L \) is the change in length, and \( L \) is the original length.
In the context of wave speed in a wire, Young's modulus allows for predicting how a wire will stretch given a certain force.
For a fixed tension and area, it determines the proportion of elongation due to thermal effects. Since mechanical properties and temperature are linked, changes in Young's modulus provide clues about thermal behavior and coefficient of linear expansion.

Thermal Expansion

Thermal expansion is the tendency of matter to change in volume in response to a change in temperature. When a material heats up, its particles begin to move faster and typically expand. The coefficient of linear expansion, \( \alpha \), quantifies this expansion:

• \( \Delta L = \alpha L \Delta T \)

where \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion, \( L \) is the original length, and \( \Delta T \) is the change in temperature.
This principle affects the wave speed in a metal wire because as the temperature increases, the length and tension of the wire change, leading to a different wave speed.
Understanding thermal expansion is crucial for determining the coefficient of linear expansion, particularly in scenarios where precise measurements and adjustments are needed, for instance, in engineering and construction.

Linear Mass Density

Linear mass density (\( \mu \)) describes how much mass is distributed along a unit length of the object, typically expressed as:

• \( \mu = \frac{\text{mass}}{\text{length}} \)

In the formula of wave speed, \( \mu \) represents the distribution of mass along the wire (\( \mu = \rho A \)). Here, \( \rho \) is the density of the material and \( A \) is the cross-sectional area.
The linear mass density influences the tension needed to achieve a particular wave speed. This is because a higher mass density requires more tension to achieve the same wave speed compared to a lesser mass density.
Calculating \( \mu \) is integral for solving problems relating to vibrations and wave propagation, ensuring that appropriate properties are maintained for desired outcomes.