91Ó°ÊÓ

A copper block is suspended in air from a wire in Part 1 of the drawing. A container of mercury is then raised up around the block as in Part \(2 .\) (a) The fundamental frequency of the wire is given by Equation 17.3 with \(n=1: f_{1}=v /(2 L)\). How is the speed \(v\) at which individual waves travel on the wire related to the tension in the wire? (b) Is the tension in the wire in Part 2 less than, greater than, or equal to the tension in Part \(1 ?\) (c) Is the fundamental frequency of the wire in Part 2 less than, greater than, or equal to the fundamental frequency in Part \(1 ?\) Justify each of your answers. In Part 2 of the drawing some \((50.0 \%)\) of the block's volume is submerged in the mercury. The density of copper is \(8890 \mathrm{~kg} / \mathrm{m}^{3},\) and the density of mercury is 13600 \(\mathrm{kg} / \mathrm{m}^{3}\). Find the ratio of the fundamental frequency of the wire in Part 2 to the fundamental frequency in Part \(1 .\) Check to see that your answer is consistent with your answers to the Concept Questions.

Step by step solution

01

Understand the relationship between wave speed and tension

The wave speed \( v \) on a string is related to the tension \( T \) in the wire by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( \mu \) is the linear mass density of the wire. This equation shows that the wave speed is proportional to the square root of the tension.

02

Analyze the tension in Part 2 compared to Part 1

In Part 1, the tension in the wire equals the weight of the copper block \( T_1 = mg \), where \( m \) is the mass of the block and \( g \) is the acceleration due to gravity. In Part 2, the block is partially submerged in mercury, experiencing a buoyant force. The tension \( T_2 = mg - F_b \), where \( F_b \) is the buoyant force. Since the buoyant force reduces the effective weight, \( T_2 < T_1 \).

03

Comparing fundamental frequencies in Part 1 and Part 2

The fundamental frequency \( f_1 = \frac{v}{2L} \), where \( v = \sqrt{\frac{T}{\mu}} \). With reduced tension in Part 2 due to buoyancy, the wave speed \( v \) decreases, so the fundamental frequency \( f_2 \) is also less than \( f_1 \). Thus, the fundamental frequency in Part 2 is less than in Part 1.

04

Calculate the buoyant force on the copper block

The buoyant force \( F_b = \rho_{Hg}V_{sub}g \), where \( \rho_{Hg} \) is the density of mercury, \( V_{sub} \) is the submerged volume (50% of block's volume), and \( g \) is gravitational acceleration. Thus, \( F_b = 0.5 \times V \times \rho_{Hg} \times g \), where \( V \) is the total volume of the block.

05

Find the mass of the block and calculate volume

The mass of the block \( m = \rho_{Cu}V \), where \( \rho_{Cu} = 8890 \, \text{kg/m}^3 \) is the density of copper. Thus, \( V = \frac{m}{\rho_{Cu}} \), giving the volume of the block.

06

Calculate the ratio of frequencies using reduced tension

With \( T_2 = mg - F_b \), the new wave speed \( v_2 = \sqrt{\frac{T_2}{\mu}} \) and old speed \( v_1 = \sqrt{\frac{T_1}{\mu}} = \sqrt{\frac{mg}{\mu}} \). The ratio \( \frac{f_2}{f_1} = \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{1 - \frac{0.5 \cdot \rho_{Hg}}{\rho_{Cu}}} \). Substitute \( \rho_{Hg} = 13600 \, \text{kg/m}^3 \) to find the frequency ratio.

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Key Concepts

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

Buoyant Force

The buoyant force is an interesting phenomenon that occurs when an object is placed in a fluid, such as mercury or water. This force acts in the opposite direction of gravity, effectively reducing the weight of the object submerged in the fluid. The buoyant force can be calculated using the formula:

• \( F_b = \rho_{fluid} \times V_{sub} \times g \)

Where:

• \( \rho_{fluid} \) is the density of the fluid.
• \( V_{sub} \) is the volume of the object that is submerged.
• \( g \) is the acceleration due to gravity.

It's important to note how the buoyant force affects the tension in a supporting wire, as seen in the example with the copper block. When part of the block is submerged, the buoyant force reduces the tension that the wire must exert, thereby influencing the wire's behavior, such as its fundamental frequency.

Wave Speed

Wave speed in a wire is a measure of how fast the wave travels through the material. It's influenced by the tension in the wire and its linear mass density \( \mu \). The wave speed \( v \) can be found using the equation:

• \( v = \sqrt{\frac{T}{\mu}} \)

Where:

• \( T \) is the tension in the wire.
• \( \mu \) is the linear mass density.

Understanding this relationship helps to explain why the wave speed decreases when the tension is reduced due to the buoyant force acting on the copper block. A lower wave speed translates to a lower fundamental frequency, as seen in the transition from Part 1 to Part 2 of the exercise.
Wave speed is crucial for understanding how energy is transmitted through materials and is a foundational concept in physics.

Linear Mass Density

Linear mass density \( \mu \) refers to the amount of mass per unit length of a material, such as a wire or string. Calculating the linear mass density is crucial for determining properties like wave speed. It is measured in kilograms per meter (kg/m).

• The formula for linear mass density is: \( \mu = \frac{m}{L} \), where \( m \) is the mass and \( L \) is the length.

In our example, knowing the linear mass density of the wire allows us to understand how mass distribution affects wave propagation.
A higher linear mass density means that the waves travel more slowly, given the same amount of tension. It has a direct correlation with how the fundamental frequency changes, especially when the tension changes due to external influences like buoyant forces.

Fundamental Frequency

The fundamental frequency of a wire or string is the lowest frequency at which it vibrates. It is a vital concept in studying wave dynamics and acoustics. The fundamental frequency \( f_1 \) is determined by the formula:

• \( f_1 = \frac{v}{2L} \)

Where:

• \( v \) is the wave speed.
• \( L \) is the length of the wire or string.

Understanding the fundamental frequency allows us to predict the sound produced by musical instruments or the behavior of engineering structures.
In situations where conditions change, like when a copper block is partially submerged in mercury, the fundamental frequency will change according to how wave speed changes. The decrease in tension from the buoyant force reduces the wave speed, thereby lowering the fundamental frequency from Part 1 to Part 2 of the problem.