Askiitians Tutor Team
Last Activity: 4 Months ago
To determine the speed of a transverse wave in a copper wire at 10 degrees Celsius, we need to consider how temperature affects the properties of the wire, specifically its tension and density. The speed of a wave in a medium is influenced by the material's Young's modulus and density, which can be adjusted for temperature changes using the coefficient of linear expansion.
Understanding Wave Speed in a Wire
The speed of a transverse wave in a wire can be calculated using the formula:
v = √(Y/ρ)
Where:
- v = speed of the wave
- Y = Young's modulus of the material
- ρ = density of the material
Adjusting for Temperature
Since the wire is taut at 30 degrees Celsius with negligible tension, we need to find the effective Young's modulus and density at 10 degrees Celsius. The Young's modulus remains relatively constant for small temperature changes, but the density will change due to thermal expansion.
The change in density can be calculated using the formula:
ρ(T) = ρ0 * (1 - α * ΔT)
Where:
- ρ0 = original density at 30 degrees Celsius
- α = coefficient of linear expansion
- ΔT = change in temperature
Calculating the New Density
Given:
- ρ0 = 9 x 103 kg/m3
- α = 1.7 x 10-5 /°C
- ΔT = 30 - 10 = 20°C
Now, substituting these values into the density formula:
ρ(10°C) = 9 x 103 kg/m3 * (1 - (1.7 x 10-5 /°C * 20°C))
ρ(10°C) = 9 x 103 kg/m3 * (1 - 0.00034)
ρ(10°C) = 9 x 103 kg/m3 * 0.99966 ≈ 8997 kg/m3
Calculating the Speed of the Wave
Now that we have the density at 10 degrees Celsius, we can use the original Young's modulus:
Y = 1.3 x 1011 N/m2
Substituting the values into the wave speed formula:
v = √(Y/ρ)
v = √(1.3 x 1011 N/m2 / 8997 kg/m3)
Calculating the fraction:
v = √(1.444 x 107 m2/s2)
v ≈ 3805.3 m/s
Final Result
Therefore, the speed of the transverse wave in the copper wire at 10 degrees Celsius is approximately 3805.3 m/s. This calculation illustrates how temperature affects the density of materials and, consequently, the speed of waves traveling through them.
