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 2.6kg 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 2.6kg 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.
Whats the diameter of the sphere?
To explain why the frequency of the fifth harmonic with the sphere submerged matches the frequency of the third harmonic before submerging the sphere, we need to understand the physics principles behind it.
When a string is fixed at both ends and oscillates, it can produce standing waves. In a standing wave, certain points on the string oscillate with maximum amplitude while others remain at rest. These points of maximum amplitude are called nodes, and the distance between adjacent nodes is known as the wavelength.
The frequency of a standing wave is determined by the rate at which nodes are formed and points on the string oscillate. The frequency can be calculated using the formula:
f = (n * v) / (2L)
where f is the frequency, n is the harmonic number, v is the speed of the wave, and L is the length of the string.
In this experiment, when the sphere is not submerged, the frequencies of the standing waves on the string correspond to the harmonics. The frequency of the third harmonic is given by:
f₁ = (3 * v) / (2L₁)
where L₁ is the length of the string without the submerged sphere.
When the sphere is completely submerged, it adds extra mass to the system, which increases the tension in the string. The increased tension affects the speed of the wave propagating along the string. We can represent this change with:
v₂ = √(T₂ / μ)
where v₂ is the speed of the wave with the submerged sphere, T₂ is the tension in the string with the sphere submerged, and μ is the linear density of the string.
Since the sphere is completely submerged, its weight is balanced by the buoyant force acting on it. This means that the tension in the string with the sphere submerged (T₂) is greater than the tension without the sphere (T₁). Therefore, v₂ is greater than v₁.
Now, let's consider the fifth harmonic with the sphere submerged. The frequency of the fifth harmonic is given by:
f₂ = (5 * v₂) / (2L₂),
where L₂ is the new length of the string with the sphere submerged.
Given that the frequency of the fifth harmonic with the sphere submerged (f₂) matches the frequency of the third harmonic before submerging the sphere (f₁), we can equate the two equations:
(5 * v₂) / (2L₂) = (3 * v₁) / (2L₁).
Since we know that v₂ > v₁, L₂ > L₁ in order to cancel out the increase in velocity. This means that the length of the string increases when the sphere is submerged.
In summary, when the 2.6kg sphere is completely submerged, the increased tension in the string affects the speed of the wave propagating along the string. This results in the length of the string increasing to maintain the same frequency relationship between harmonics.