The density of copper is 9.8 kilograms per meter cube and the density of gold is 1.9 kilograms per meter cube. When two wires of these metals are held under the same tension, the wave speed in the gold is found to be half that in the copper wire. What is the ratio of the diameters of the two wires?
To find the ratio of the diameters of the two wires, we need to use the relationship between the wave speed, tension, and density of a wire.
The wave speed in a wire is given by the equation:
v = sqrt(T/μ)
where v is the wave speed, T is the tension, and μ is the linear mass density of the wire (mass per unit length).
We are given that the wave speed in the gold wire (v_gold) is half of that in the copper wire (v_copper). Therefore, we can write:
v_gold = 0.5 * v_copper
We can also write the ratio of the linear mass densities as:
μ_gold / μ_copper = ρ_gold / ρ_copper
where μ_gold and μ_copper are the linear mass densities of the gold and copper wires, respectively, and ρ_gold and ρ_copper are the densities of gold and copper, respectively.
Substituting the given densities, we have:
μ_gold / μ_copper = (1.9 kg/m^3) / (9.8 kg/m^3)
Now, let's use the relationship between linear mass density and the cross-sectional area (A) of a wire:
μ = m/A
where μ is the linear mass density, m is the mass of the wire, and A is the cross-sectional area.
Since the wires are under the same tension, the tension cancels out in the equation. Therefore, we have:
μ_gold / μ_copper = (m_gold / A_gold) / (m_copper / A_copper)
Now, let's assume that the lengths of the two wires are the same, so their masses are directly proportional to their volumes.
Since density is mass per unit volume, we can write:
m_gold / V_gold = ρ_gold
m_copper / V_copper = ρ_copper
where V_gold and V_copper are the volumes of the gold and copper wires, respectively.
So, we have:
μ_gold / μ_copper = ρ_gold / ρ_copper = (m_gold / V_gold) / (m_copper / V_copper)
Since the masses of the wires are directly proportional to their volumes, we can write:
m_gold / m_copper = V_gold / V_copper
Substituting this into the previous equation, we have:
μ_gold / μ_copper = V_gold / V_copper
Now, let's consider the relationship between the diameter (d) and the cross-sectional area (A) of a wire:
A = π * (d^2 / 4)
Substituting this into the previous equation, we have:
μ_gold / μ_copper = (π * (d_gold^2 / 4)) / (π * (d_copper^2 / 4))
Canceling out the π and the 4, we get:
μ_gold / μ_copper = (d_gold^2) / (d_copper^2)
Rearranging this equation, we obtain:
(d_gold^2) / (d_copper^2) = μ_gold / μ_copper
Taking the square root of both sides:
d_gold / d_copper = sqrt(μ_gold / μ_copper)
Now, substitute the given densities:
d_gold / d_copper = sqrt((1.9 kg/m^3) / (9.8 kg/m^3))
Calculating this ratio, we find:
d_gold / d_copper ≈ 0.448
Therefore, the ratio of the diameters of the two wires is approximately 0.448.
To solve this problem, we can use the formula for wave speed, which is given by:
Wave speed = √(Tension / (Linear density * π * radius^2))
Where:
- Wave speed is the speed at which waves propagate through the wire.
- Tension is the force applied to the wire.
- Linear density is the mass per unit length of the wire.
- π is the mathematical constant pi.
- Radius is the radius of the wire.
Let's assume that the tension, force, and π are the same for both copper and gold wires. We can then set up the following equations:
Wave speed in copper = √(Tension / (Linear density of copper * π * radius of copper^2))
Wave speed in gold = √(Tension / (Linear density of gold * π * radius of gold^2))
We are given that the wave speed in gold is half that in the copper wire, so we can write:
Wave speed in gold = 0.5 * Wave speed in copper
Substituting the formulas for wave speed into the equation above, we get:
√(Tension / (Linear density of gold * π * radius of gold^2)) = 0.5 * √(Tension / (Linear density of copper * π * radius of copper^2))
Simplifying the equation by canceling out the common terms:
√(Linear density of copper / Linear density of gold) * √(radius of copper^2 / radius of gold^2) = 0.5
Taking the square of both sides allows us to remove the square root:
(Linear density of copper / Linear density of gold) * (radius of copper^2 / radius of gold^2) = 0.5^2
Substituting the given values for the linear density of copper and gold:
(9.8 / 1.9) * (radius of copper^2 / radius of gold^2) = 0.5^2
Simplifying the equation further:
5.15789474 * (radius of copper^2 / radius of gold^2) = 0.25
Dividing both sides by 5.15789474:
(radius of copper^2 / radius of gold^2) = 0.25 / 5.15789474
(radius of copper^2 / radius of gold^2) ≈ 0.04852071006
Taking the square root of both sides:
(radius of copper / radius of gold) ≈sqrt(0.04852071006)
(radius of copper / radius of gold) ≈ 0.2203941472
Therefore, the ratio of the diameters of the two wires is approximately 0.2204.