In a laboratory experiment, one end of a horizontal string is tied to a support while the other end passes over a frictionless pulley and is tied to a 1.5 kg sphere. Students determine the frequencies of standing waves on the horizontal segment of the string, then they raise a beaker of water until the hanging 1.5 kg sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged exactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?

Verified step by step guidance

1. Step 1: Understand the problem. The frequency of standing waves on a string depends on the tension in the string and the linear mass density. The tension in the string is provided by the weight of the sphere. When the sphere is submerged in water, the buoyant force reduces the effective weight of the sphere, thereby reducing the tension in the string. The problem involves matching the frequencies of the fifth harmonic (with the sphere submerged) and the third harmonic (before submersion).
2. Step 2: Write the formula for the frequency of standing waves on a string. The frequency of the nth harmonic is given by: $f=\frac{n}{2}\surd \frac{T}{\mu }$ where $n$ is the harmonic number, $T$ is the tension in the string, and $\mu$ is the linear mass density of the string. Note that the linear mass density remains constant throughout the problem.
3. Step 3: Relate the tension in the string to the weight of the sphere. Before submersion, the tension is equal to the weight of the sphere: $T=mg$ where $m$ is the mass of the sphere (1.5 kg) and $g$ is the acceleration due to gravity. When the sphere is submerged, the tension is reduced by the buoyant force: $T=mg-\rho Vg$ where $\rho$ is the density of water and $V$ is the volume of the sphere.
4. Step 4: Use the condition that the fifth harmonic frequency with the sphere submerged matches the third harmonic frequency before submersion. Equate the two frequencies: $\frac{5}{2}\surd \frac{T}{\mu }=\frac{3}{2}\surd \frac{T}{\mu }$ Substitute the expressions for tension before and after submersion into this equation and simplify to solve for the volume $V$ of the sphere.
5. Step 5: Relate the volume of the sphere to its diameter. The volume of a sphere is given by: $V=\frac{4}{3}\pi (\frac{d}{2}{)}^3$ where $d$ is the diameter of the sphere. Solve for $d$ using the volume obtained in the previous step.

Key Concepts

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In a string fixed at both ends, these waves create nodes (points of no displacement) and antinodes (points of maximum displacement). The frequency of standing waves is determined by the length of the string and the tension in it, leading to specific harmonics that can be observed in experiments.

Harmonics

Harmonics are specific frequencies at which a system can oscillate, resulting in standing waves. The fundamental frequency is the first harmonic, while higher harmonics are integer multiples of this frequency. In the context of the string, the third harmonic corresponds to a specific pattern of nodes and antinodes, and understanding the relationship between different harmonics is crucial for analyzing the effects of changes in the system, such as submerging the sphere.

Buoyancy and Displacement

Buoyancy is the upward force exerted by a fluid on an object submerged in it, which is equal to the weight of the fluid displaced by the object. When the sphere is submerged, it displaces a volume of water equal to its own volume, affecting the tension in the string and consequently the frequency of the standing waves. The relationship between the submerged object's volume and the frequency of the harmonics is key to determining the sphere's diameter.