2. The difference in length between a
brass and an iron rod is 14 cm at
10 C. ° What must be the length of
the iron for this difference to remain
14 cm, when both rods are heated
to 100 C° ? (Linear expansivity of
brass 6 19 10 / K − = × and for iron 6 12 10 / K).
To solve this problem, we can use the formula for linear expansion:
ΔL = αLΔT
Where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
Let's assume the original length of the brass rod is L_b and the original length of the iron rod is L_i.
Given that the difference in length between the two rods at 10°C is 14 cm, we can write the equation:
(L_i - L_b) = α_brass * L_b * (100 - 10)
We know that α_brass is equal to 6.19 * 10^-6 / °C. Solving for L_b, we get:
L_b = (L_i - 14) / (1 + 6.19 * 10^-6 * 90)
Now, we need to find the length of the iron rod at 100°C. We can use the same equation and solve for L_i:
(L_i - L_b) = α_iron * L_i * (100 - 10)
Given that α_iron is equal to 6.12 * 10^-6 / °C, we can substitute in the value of L_b from the previous equation. Solving for L_i, we get:
L_i = L_b / (1 - 6.12 * 10^-6 * 90)
Substituting the values we know:
L_b = (L_i - 14) / (1 + 6.19 * 10^-6 * 90)
L_i = L_b / (1 - 6.12 * 10^-6 * 90)
We can substitute L_b into the second equation to solve for L_i:
L_i = (L_i - 14) / (1 + 6.19 * 10^-6 * 90) / (1 - 6.12 * 10^-6 * 90)
Solving this equation will give us the length of the iron rod when both rods are heated to 100°C.