The active element of a certain laser is an ordinary glass rod 19.5 cm long and 1.04 cm in diameter. If the temperature of the rod increases by 77.3°C, calculate its increase in length.Calculate its increase in volume. Calculate its increase in diameter. (Use 9.0×10-6 °C-1 for the coefficient of linear expansion for glass. Use this value to find the coefficient of volume expansion in the later parts of the problem.)
To find the increase in length of the glass rod, we can use the formula:
ΔL = αL₀ΔT
Where:
ΔL = increase in length
α = coefficient of linear expansion
L₀ = initial length of the rod
ΔT = change in temperature
Given:
α = 9.0×10^(-6) °C^(-1)
L₀ = 19.5 cm
ΔT = 77.3°C
Substituting the values into the formula, we have:
ΔL = (9.0×10^(-6) °C^(-1))(19.5 cm)(77.3°C)
Calculating this, we find:
ΔL ≈ 0.01335 cm
Therefore, the increase in length of the glass rod is approximately 0.01335 cm.
To find the increase in volume, we can use the formula relating volume expansion to linear expansion:
ΔV = βV₀ΔT
Where:
ΔV = increase in volume
β = coefficient of volume expansion (which can be related to the coefficient of linear expansion through β = 3α)
V₀ = initial volume of the rod (which can be calculated using V₀ = πr₀²L₀, where r₀ is the initial radius)
ΔT = change in temperature
Given the information about the rod's dimensions, we can calculate its initial volume as follows:
V₀ = πr₀²L₀
Given: r₀ = 0.52 cm (half the diameter of the rod)
Substituting the values into the equation, we have:
V₀ = π(0.52 cm)²(19.5 cm)
Calculating this, we find:
V₀ ≈ 5.180 cm³
Since β = 3α, we can substitute β = 3(9.0×10^(-6)) into the formula for ΔV to find:
ΔV = (3α)V₀ΔT
ΔV = (3)(9.0×10^(-6))(5.180 cm³)(77.3°C)
Calculating this, we find:
ΔV ≈ 0.0128 cm³
Therefore, the increase in volume of the glass rod is approximately 0.0128 cm³.
To calculate the increase in diameter, we need to consider that the diameter is twice the radius. Thus, the increase in diameter (Δd) is twice the increase in radius (Δr):
Δd = 2Δr
Since radius (r) is half the diameter, we can use the formula Δr = αr₀ΔT to find:
Δr = (9.0×10^(-6))(0.52 cm)(77.3°C)
Calculating this, we find:
Δr ≈ 0.03618 cm
Finally, substituting this value into the formula for Δd, we find:
Δd = 2(0.03618 cm)
Calculating this, we find:
Δd ≈ 0.07236 cm
Therefore, the increase in diameter of the glass rod is approximately 0.07236 cm.