steeel and silver wire of the same diameter and same length are stretched with equal tension. what is the fundamental frequency of the silver wire if that of the steel is 200 Hz.
To find the fundamental frequency of the silver wire, we can use the formula for the fundamental frequency of a vibrating string:
f = (1/2L) √(T/μ)
Where:
f = fundamental frequency
L = length of the wire
T = tension in the wire
μ = linear mass density of the wire
Given that the steel wire has a fundamental frequency of 200 Hz, we can assume that the tension and length are the same for both wires. Therefore, we only need to compare the linear mass density (μ) of the two wires.
The linear mass density (μ) of a wire is equal to the mass (m) divided by the length (L):
μ = m/L
Since the two wires have the same diameter and length, their mass will be directly proportional to their material densities.
Hence, if the density of steel is ρ_steel and the density of silver is ρ_silver, and the wires have the same length and diameter, then:
μ_steel = ρ_steel/L
μ_silver = ρ_silver/L
Now, we're given that the wires have the same tension. Since tension is directly related to the material density and length, the linear mass density (μ) and tension (T) for both wires will be the same.
Thus, we can write:
μ_steel/T = μ_silver/T
Since the tension cancels out on both sides, it implies:
μ_steel = μ_silver
Therefore, the linear mass density of the silver wire is the same as that of the steel wire.
Consequently, the fundamental frequency of the silver wire will also be 200 Hz, the same as the steel wire.
To find the fundamental frequency of the silver wire, we can use the formula for the fundamental frequency of a stretched string:
f = (1/2L) * √(T/μ)
Where:
f is the frequency (in Hz)
L is the length of the wire
T is the tension applied to the wire
μ is the linear mass density of the wire
Given that the silver wire has the same diameter and length as the steel wire and is stretched with equal tension, the only difference between the two wires is the linear mass density (μ).
Since the linear mass density is inversely proportional to the square of the diameter, and the diameter is the same for both wires, the linear mass density of the silver wire will be greater than that of the steel wire.
Let's assume the linear mass density of the steel wire to be μs, and the linear mass density of the silver wire to be μAg.
Since the silver wire is denser than the steel wire, we have the relationship: μAg > μs
Now, let's substitute these values into the formula for the fundamental frequency of the silver wire:
= (1/2L) * √(T/μAg)
Since the tension (T) and the length (L) are the same for both wires, we can simplify the equation as follows:
= √(μs/μAg) * fsteel
Given that the fundamental frequency of the steel wire (fsteel) is 200 Hz, we can substitute this value into the equation:
= √(μs/μAg) * 200 Hz
Now, we want to find the fundamental frequency of the silver wire, so we rearrange the equation to solve for :
= (√(μs/μAg)) * 200 Hz
By solving this equation, you can find the fundamental frequency of the silver wire.
The fundamental frequency will be proportional to the transverse wave speed.
The silver/steel frequency ratio equals sqrt(steel density/silver density) = sqrt(7.7/10.5) = 0.85
That makes the fundamental frequency for the silver wire about 170 Hz.