1. A solid copper cylinder hangs at the bottom of a steel wire of negligible mass. The top end of the wire is fixed. When the wire is struck, it emits sound with a fundamental frequency of 300 Hz. The copper cylinder is then submerged in water so that half its volume is below the water line. Determine the new fundamental frequency
This question was answered by me last week. See "related questions" below.
To determine the new fundamental frequency, we need to consider the change in effective length and effective speed of sound in the wire due to the presence of water.
1. Start with the fundamental frequency equation for a stretched string:
f = (1 / 2L) * sqrt(T / μ),
where f is the fundamental frequency, L is the length of the wire, T is the tension in the wire, and μ is the linear density of the wire.
2. Since the wire is struck, the tension and linear density remain constant, so the equation simplifies to:
f ∝ 1 / L.
3. When the cylinder is submerged in water, half its volume is below the water line. This means that the effective length of the wire has increased by half the length of the cylinder.
4. Let's denote the original length of the wire as L0 and the length of the cylinder as hc. The new effective length of the wire will be L = L0 + hc/2.
5. To determine the new fundamental frequency, we need to find the ratio of the new frequency (f') to the original frequency (f):
f' / f = L0 / L.
6. Substituting the expression for L:
f' / f = L0 / (L0 + hc/2).
7. Now, let's consider the change in the effective speed of sound in the wire due to the presence of water. The speed of sound in a medium is given by:
v = sqrt(T / μ).
8. When the copper cylinder is submerged in water, it will displace some water, causing an increase in the effective mass per unit length of the wire. Let's denote the original effective speed of sound as v0 and the change in the effective speed of sound as Δv.
9. The new effective speed of sound in the wire will be v = v0 + Δv.
10. The change in speed of sound Δv can be calculated using the relationship between the bulk modulus and speed of sound in a medium:
Δv / v0 = - ΔV / V,
where ΔV is the volume change and V is the original volume occupied by the material.
11. Since half the volume of the copper cylinder is submerged, ΔV = V/2.
12. Substituting the expression for V:
Δv / v0 = - (hc/2) / V,
Δv / v0 = - (hc/2) / (hc + V0),
where V0 is the original volume of the copper cylinder.
13. Now, let's consider the change in the tension in the wire due to the presence of water. The tension in the wire will be slightly increased by the weight of the submerged part of the copper cylinder.
14. The change in tension ΔT can be calculated using the relationship between the buoyant force and weight:
ΔT = mg,
where m is the mass of the water displaced by the submerged part of the copper cylinder and g is the acceleration due to gravity.
15. The mass of water displaced is equal to the mass of the submerged part of the copper cylinder:
m = ρ_water * V / 2,
where ρ_water is the density of water.
16. Substituting the expressions for m and V:
ΔT = ρ_water * V0 * g / 2.
17. Finally, substitute the expressions for Δv and ΔT in the equation for the change in fundamental frequency (f' / f):
f' / f = (L0 / (L0 + hc/2)) * ((v0 + Δv) / v0) * sqrt((T + ΔT) / (T)).
We then simplify this equation to find the new fundamental frequency in terms of the original frequency and known quantities.
To determine the new fundamental frequency after the copper cylinder is submerged in water, we need to consider the change in effective length of the steel wire due to the change in mass of the cylinder.
Let's break down the problem step by step:
1. Start with the original scenario, where the copper cylinder is in air and the wire is fixed at the top. In this case, the fundamental frequency is 300 Hz.
2. When the wire is struck, it vibrates with a certain length to produce this frequency. We can calculate this length using the formula for the wavelength of a standing wave:
λ = 2L
where λ is the wavelength and L is the length of the vibrating wire.
3. Now, let's consider what happens when the copper cylinder is submerged in water. When an object is submerged in a fluid, it experiences a buoyant force that opposes its weight. In this case, half of the copper cylinder's volume is below the water line, so half of its weight is supported by the buoyant force.
This means that the effective mass of the copper cylinder hanging from the wire is reduced by half.
4. Since the mass of the copper cylinder affects the tension in the wire, which in turn affects the speed of sound propagation, the effective length of the wire will change.
5. To find the new effective length, we need to find the ratio of the original mass to the reduced mass:
original mass / reduced mass = original length / new effective length
Since the original mass is twice the reduced mass, the new effective length will be half the original length.
So, the effective length of the wire is halved when the copper cylinder is submerged.
6. Now that we know the new effective length, we can calculate the new fundamental frequency using the same formula:
λ = 2L
Since the effective length is halved, the new wavelength will be halved as well:
λ_new = λ_original / 2
Therefore, the new fundamental frequency can be calculated as:
f_new = (speed of sound) / λ_new
However, in this case, we are not given the speed of sound, so we cannot provide an exact value for the new fundamental frequency. If we assume the speed of sound in air is 343 m/s (at standard temperature and pressure), we can estimate the new fundamental frequency using the given value for the original frequency of 300 Hz:
f_new = (speed of sound) / λ_new
f_new = 343 m/s / (λ_original / 2)
f_new = 343 m/s / (2L_original)
f_new = 171.5 m/s / L_original
f_new = 171.5 m/s / (original length of the wire)
We would need to know the original length of the wire in order to calculate the new fundamental frequency accurately.