When the temperature of a thin silver [α = 19 × 10-6 (C°)-1] rod is increased, the length of the rod increases by 3.6 × 10-3 cm. Another rod is identical in all respects, except that it is made from gold [α = 14 × 10-6 (C°)-1]. By how much ΔL does the length of the gold rod increase when its temperature increases by the same amount as that for the silver rod?
(14*10^-6/19*10^-6) * 3600cm = 2653 cm.
To find the change in length (ΔL) of the gold rod, we can use the formula:
ΔL = α * L * ΔT
where:
ΔL is the change in length
α is the coefficient of linear expansion
L is the original length of the rod
ΔT is the change in temperature
First, we need to determine the original length of the gold rod. Since it is identical to the silver rod, we can use the change in length of the silver rod to find the original length:
ΔL_silver = α_silver * L * ΔT
ΔL_silver = (19 × 10^-6 C⁻¹) * L * ΔT
We can rearrange this equation to solve for L:
L = ΔL_silver / (α_silver * ΔT)
Now we can substitute the given values:
L = (3.6 × 10^-3 cm) / (19 × 10^-6 C⁻¹ * ΔT)
L = 189.5 cm / ΔT
Now that we know the original length (L) of the gold rod, we can calculate the change in length (ΔL_gold) using the same formula:
ΔL_gold = α_gold * L * ΔT
Substituting the given values and the calculated value for L:
ΔL_gold = (14 × 10^-6 C⁻¹) * (189.5 cm / ΔT) * ΔT
The ΔT cancels out, so:
ΔL_gold = 14 × 10^-6 C⁻¹ * 189.5 cm
ΔL_gold ≈ 0.0027 cm
Therefore, the length of the gold rod increases by approximately 0.0027 cm when its temperature increases by the same amount as the silver rod.