The active element of a certain laser is made of a glass rod 29.0 cm long and 1.20 cm in diameter. Assume the average coefficient of linear expansion of the glass is equal to 9.00 10-6 (°C)-1. The temperature of the rod increases by 70.0°C.
(a) What is the increase in its length?
(b) What is the increase in its diameter?
(c) What is the increase in its volume?
To find the increase in length of the glass rod, we need to use the formula for linear expansion:
ΔL = α * L₀ * ΔT
where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the initial length, and ΔT is the change in temperature.
(a) First, let's find the increase in length using the given values:
ΔL = (9.00 * 10^(-6) (°C)^(-1)) * (29.0 cm) * (70.0°C)
Calculating the expression:
ΔL = 0.000009 * 29 * 70 = 0.0189 cm
Therefore, the increase in length of the glass rod is 0.0189 cm.
To find the increase in diameter of the glass rod, we need to use the formula for linear expansion:
ΔD = 2 * α * D₀ * ΔT
where ΔD is the change in diameter, α is the coefficient of linear expansion, D₀ is the initial diameter, and ΔT is the change in temperature.
(b) Let's calculate the increase in diameter using the given values:
ΔD = 2 * (9.00 * 10^(-6) (°C)^(-1)) * (1.20 cm) * (70.0°C)
Calculating the expression:
ΔD = 2 * 0.000009 * 1.20 * 70 = 0.001512 cm
Therefore, the increase in diameter of the glass rod is 0.001512 cm.
To find the increase in volume of the glass rod, we need to use the formula for volume expansion:
ΔV = β * V₀ * ΔT
where ΔV is the change in volume, β is the coefficient of volume expansion, V₀ is the initial volume, and ΔT is the change in temperature.
Since the increase in volume is not given, we will first calculate the initial volume of the glass rod:
V₀ = π * r² * L₀
where V₀ is the initial volume, r is the radius, and L₀ is the initial length.
We know that the diameter is given as 1.20 cm, so the radius will be half of that:
r = 0.60 cm = 0.0060 m
Substituting the values into the equation:
V₀ = π * (0.0060 m)² * (0.29 m)
Calculating the expression:
V₀ = 0.094416 m³
Now we can calculate the increase in volume using the given values:
ΔV = (β) * (V₀) * (ΔT)
Given that β ≈ 3α for most materials:
ΔV ≈ (3α) * (V₀) * (ΔT)
Calculating the expression:
ΔV = (3 * 9.00 * 10^(-6) (°C)^(-1)) * (0.094416 m³) * (70.0°C)
ΔV ≈ 0.019707 m³
Therefore, the increase in volume of the glass rod is approximately 0.019707 m³.