Physics 2010 JAMB Past Questions - Question 16: Two metals P and Q of lengths are heated through the same temperature difference. if the ratio of linear expansivities of P to Q is 2:3 and the ratio of their lengths is 3:4. What is the ratio of increase in lengths of P to Q?

Correct Answer

C

Explanation

The increase in length of a material due to heating can be calculated using the formula:

ΔL = α * L * ΔT

Where:
ΔL = Change in length
α = Linear expansivity
L = Original length
ΔT = Change in temperature

Let's assume that the increase in length for metal P is ΔLP, and the increase in length for metal Q is ΔLQ.

Given that the ratio of linear expansivities of P to Q is 2:3, we can write:

αP/αQ = 2/3

Given that the ratio of their lengths is 3:4, we can write:

LP/LQ = 3/4

We want to find the ratio of ΔLP to ΔLQ.

We can rearrange the linear expansivity ratio equation to express αP in terms of αQ:

αP = (2/3)αQ

Now, let's calculate the ratio of ΔLP to ΔLQ:

(ΔLP/ΔLQ) = (αP * LP * ΔT) / (αQ * LQ * ΔT)

Since ΔT is the same for both metals (heated through the same temperature difference), it cancels out:

(ΔLP/ΔLQ) = (αP * LP) / (αQ * LQ)

Substitute the expression for αP from the linear expansivity ratio equation:

(ΔLP/ΔLQ) = ((2/3)αQ * LP) / (αQ * LQ)

Now, simplify:

(ΔLP/ΔLQ) = (2/3) * (LP/LQ)

We know that LP/LQ = 3/4, so:

(ΔLP/ΔLQ) = (2/3) * (3/4)

Now, calculate the ratio:

(ΔLP/ΔLQ) = (2/3) * (3/4) = 2/4 = 1/2

So, the ratio of the increase in length of metal P to metal Q is 1:2.