A solid copper object 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 a sound with a fundamental frequency of 135 Hz. The copper object is then submerged in water so that half its volume is below the water line. Determine the new fundamental frequency.
Hz
Density of copper= 8960 kg/m^3
" " "" water = 1000 kg/m^3
v= √(T/ρ)= λf
λ= 1/f1 √(T1/ρ)= 1/f2 √(T2/ρ)
1 is before submersion and 2 is after
f2= f1 √(T2/T1)
T2= T1-B
B is the buoyant force
B= ρ(w)V(w)g
(w) denotes the density and volume of water, otherwise I'll be referring to the hanging mass
V(w)= V/2
Since half of the hanging object is submerged
V= m/ρ
Put it all together
f2= f1 √(T2/T1)= f1 √((T1-B)/T1)
= f1 √(1-(B/T1))
= f1 √(1-((ρ(w)(V/2)g)/mg))
= f1 √(1-((ρ(w)(m/ρ)g)/2mg))
Cancel the m and g
= f1 √(1-(p(w)/2ρ))
= 135√(1-(1000/2(8960)))
= 131.2 Hz
Hope that helps
To determine the new fundamental frequency, we need to consider the change in effective length and density of the wire.
1. Effective Length:
When the copper object is submerged in water, its buoyancy force reduces the tension in the wire. As a result, the wire becomes longer, and the effective length increases.
2. Density:
The density of the wire changes when the copper object is submerged, as its weight is partially supported by the buoyant force of water. Therefore, the effective density of the wire decreases.
To calculate the new fundamental frequency, we need to consider how these changes affect the wave speed in the wire.
The wave speed in a stretched wire is given by the equation:
v = √(T/μ)
where v is the speed of the wave, T is the tension in the wire, and μ is the linear density (mass per unit length).
The fundamental frequency (f) of a standing wave on the wire is related to the wave speed (v) and effective length (L) by the equation:
f = v / (2L)
Let's calculate the changes in effective length and density.
1. Effective Length:
If the wire length before the copper object was submerged was L1, then the effective length after submerging is L2 = L1 + δL, where δL is the increase in length due to the reduced tension.
2. Density:
The density of the wire after submerging is given by:
μ2 = (μ1 * V1 + ρw * Vw) / (V1 + Vw)
where μ2 is the new density of the wire, μ1 is the original density of the wire, V1 is the volume of wire above the water line, ρw is the density of water, and Vw is the volume of water displaced by the submerged part of the copper object.
Now, we can calculate the new fundamental frequency:
f2 = v2 / (2 * L2)
Simplifying the equation, we have:
f2 = [√(T/μ2)] / [2 * (L1 + δL)]
The tension (T) in the wire remains the same, as it is not affected by the copper object being submerged.
In order to calculate the new fundamental frequency, we need the numerical values for the variables T, μ1, V1, ρw, and Vw.
To determine the new fundamental frequency, we need to consider the change in effective length of the wire after the copper object is submerged in water.
1. The fundamental frequency of a vibrating string or wire is inversely proportional to its effective length. When the copper object is at the bottom, the effective length of the wire is the distance between the top end (fixed) and the bottom end (where the copper object hangs).
2. When half of the copper object's volume is submerged in water, it displaces an equal volume of water. According to Archimedes' principle, the volume of water displaced is equal to the volume of the object submerged.
3. Since density = mass/volume, the density of the copper object is greater than that of water. Therefore, when submerged, it will experience a buoyant force that partially counteracts its weight. This results in a reduction in the effective length of the wire.
To determine the new fundamental frequency, we can use the equation:
f = (1/2L) * sqrt(T/μ),
where:
- f is the fundamental frequency,
- L is the effective length of the wire,
- T is the tension in the wire, and
- μ is the linear mass density of the wire.
Since the tension and linear mass density of the wire remain constant, we only need to calculate the change in effective length (L).
As the copper object is submerged by half its volume, we can assume it displaces an equal volume of water. The volume of water displaced is equal to the volume of the object submerged.
Now, let's assume that the initial length of the copper object hanging below the water line is L1. After half its volume is submerged, the length of the submerged part of the copper object is L1/2.
Therefore, the change in effective length is equal to the length of the submerged part of the copper object, i.e., L1/2.
The new effective length of the wire is the original length minus the change in effective length.
Hence, the new effective length is L - L1/2.
Finally, we can substitute the new effective length into the fundamental frequency formula to calculate the new fundamental frequency (f'):
f' = (1/2(L - L1/2)) * sqrt(T/μ).
Note: To obtain the specific values for the equation above, you would need to know the length of the wire, the length of the copper object below the water line, the tension in the wire, and the linear mass density of the wire.