Problem 81 - University Physics with Modern Physics

Question

A large rock that weighs 164.0 \(\mathrm{N}\) is suspended from the lower end of a thin wire that is 3.00 \(\mathrm{m}\) long. The density of the rock is 3200 \(\mathrm{kg} / \mathrm{m}^{3} .\) The mass of the wire is small enough that its effect on the tension in the wire can be neglected. The upper end of the wire is held fixed. When the rock is in air, the fundamental frequency for transverse standing waves on the wire is 42.0 Hz. When the rock is totally submerged in a liquid, with the top of the rock just below the surface, the fundamental frequency for the wire is 28.0 Hz. What is the density of the liquid?

Step-by-Step Solution

Answer

The density of the liquid is approximately 1300 kg/m³.

Step 1: Determine the volume of the rock

To find the volume of the rock, use the formula for volume based on density: \( V = \frac{m}{\rho} \), where \( m \) is the mass and \( \rho \) is the density. The weight of the rock is 164.0 N, and using \( F = mg \), we find the mass: \( m = \frac{F}{g} = \frac{164.0 \text{ N}}{9.8 \text{ m/s}^2} \). Substitute \( m \) and \( \rho = 3200 \text{ kg/m}^3 \) to find \( V \).

Step 2: Calculate the mass of the displaced liquid

Use Archimedes' principle, which states that the buoyant force is equal to the weight of the liquid displaced: \( F_b = \rho_{liquid} V g \). When submerged, the tension in the wire changes. Let \( T_{air} \) and \( T_{liquid} \) be the tensions. The change in tension is due to the buoyant force, so \( T_{liquid} = T_{air} - F_b \). We will use the frequencies to find these tensions.

Step 3: Relate frequency to tension

The fundamental frequency of a string is given by \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( L \) is the length of the wire, \( T \) is the tension, and \( \mu \) is the linear density of the wire. Since the mass of the wire is negligible, \( \mu \) can be considered constant. We have two frequencies: \( f_{air} = 42.0 \) Hz and \( f_{liquid} = 28.0 \) Hz. From these frequencies, set up equations using \( T_{air} \) and \( T_{liquid} \).

Step 4: Solve for liquid's density

Using the relationship between tension and frequency from the previous step, we know: \( T_{air} = (2Lf_{air})^2 \mu \) and \( T_{liquid} = (2Lf_{liquid})^2 \mu \). Also, \( T_{liquid} = T_{air} - \rho_{liquid} V g \). Arrange these equations to solve for \( \rho_{liquid} \). Substitute the values we know to find \( \rho_{liquid} \).

Key Concepts

Archimedes' Principle

Archimedes' principle is a fundamental theory of fluid mechanics. It explains why objects seem lighter when submerged in a liquid. The principle states that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces.
For example, when the rock in the exercise is submerged in the liquid, it displaces a certain volume of liquid. According to Archimedes' principle, the buoyant force on the rock is equal to the weight of that displaced liquid. This reduces the effective tension in the wire compared to when the rock is in the air, as part of its weight is "supported" by the buoyant force.

• Buoyant Force: It's the force that a fluid exerts on an object that is partially or fully submerged.
• Displaced Fluid: It's the amount of fluid that is pushed aside to make space for the submerged object.
• Application: This principle is used to calculate the density of liquids by observing the change in tension caused by submersion.

Understanding Archimedes' principle helps us in various fields, from engineering to understanding basic physics principles.

Transverse Standing Waves

Transverse standing waves are waves where the medium moves perpendicular to the direction of the wave. In the context of the thin wire holding a rock, these waves are created along the length of the wire.
When the wire vibrates transversely due to a rock hanging at its end, standing waves can form. These have specific patterns where some points (nodes) do not move, and others (antinodes) oscillate with maximum amplitude.

• Standing Wave Pattern: At particular frequencies, specific parts of the wire (nodes) remain stationary while others (antinodes) move.
• Parameters Affecting Standing Waves: The tension in the wire and the wire's length primarily determine the wave patterns and frequencies.
• Practical Implications: Determining the frequency of these waves helps us understand the tension changes caused by submerging the rock in a liquid.

By analyzing standing waves, we gain insight into the physical characteristics and conditions of the system, such as the wire or the submerged object.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system resonates, producing a standing wave. For a wire, this is the primary harmonic where only a single half-wavelength fits along the wire's length.
In practical terms, the fundamental frequency of the wire changes based on tension. When the rock is in the air, the tension and thus the frequency are higher compared to when the rock is submerged due to reduced tension from buoyancy.

• Relationship with Tension: The fundamental frequency directly relates to the tension in the wire; more tension results in a higher frequency.
• Measuring Changes: By calculating the difference in frequency when the rock is in air versus submerged, one can deduce changes in tension and thus determine liquid properties like density.
• Importance: Understanding and calculating fundamental frequencies help solve practical problems, such as determining fluid density in experimental setups.

These insights are critical not only in academic exercises but also in real-world applications, such as musical instruments or engineering solutions.