Two wires made of the same material of density 7240 kg/m3 and having the radius in the ratio 1:2 but the same length 1.54 connected together and stretched by a force of 540 N. If the speed of a transverse wave through the thinner wire is 3600 m/s, calculate the wave velocity in the thicker wire.
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To calculate the wave velocity in the thicker wire, we need to use the formula:
v = √(T/μ)
where v is the wave velocity, T is the tension in the wire, and μ is the linear density of the wire.
First, let's determine the linear density of the wires. The linear density (μ) is given by the formula:
μ = m/L
where m is the mass of the wire and L is its length.
Since the wires are made of the same material and have the same length, their linear densities will be the same.
To calculate the mass of each wire, we can use the formula:
m = ρ * V
where m is the mass, ρ is the density of the wire, and V is the volume of the wire.
The volume of a wire is given by the formula:
V = π * r^2 * L
where r is the radius of the wire and L is its length.
Let's denote the radius of the thinner wire as r and the radius of the thicker wire as 2r, as given in the problem.
Since both wires have the same length, their volumes will be proportional to the square of their radii:
V_thinner / V_thicker = (π * r^2 * L) / (π * (2r)^2 * L)
V_thinner / V_thicker = r^2 / (4r^2)
V_thinner / V_thicker = 1/4
Therefore, the volume of the thicker wire is four times the volume of the thinner wire.
Now, let's calculate the mass of the thinner wire:
m_thinner = ρ * V_thinner
And the mass of the thicker wire:
m_thicker = ρ * V_thicker = 4 * ρ * V_thinner
Since the tension in the wires is the same, we can set up an equation using the wave velocity formula:
v_thinner = √(T / μ_thinner)
And
v_thicker = √(T / μ_thicker)
Squaring both sides of the equation, we get:
v_thinner^2 = T / μ_thinner
And
v_thicker^2 = T / μ_thicker
Since we know that μ_thicker = 4 * μ_thinner, we can substitute it into the equation:
v_thicker^2 = T / (4 * μ_thinner)
Now we can calculate the wave velocity in the thicker wire by substituting the known values:
v_thicker = √(540 / (4 * μ_thinner))
To find the value of μ_thinner, we need to know the density of the material. You mentioned that the density is 7240 kg/m^3.
Now we can substitute the value of μ_thinner into the equation:
v_thicker = √(540 / (4 * (7240 * (π * r^2 * L))))
Simplifying the equation further will give us the final answer for the wave velocity in the thicker wire.