2018 A/L Physics Past Paper · Question 07 (English Medium)

Answer: (5) $2.4 \times 10^{-7}\ ^\circ\text{C}^{-1}$

Given information

We know: the fractional change in length is $\Delta l/l = 2.4 \times 10^{-5}$. It has no unit.

The temperature rises by $\Delta\theta = 100\ ^\circ\text{C}$.

What the question asks

We want: the linear expansivity $\alpha$ of the material of the rod. Its unit is ${}^{\circ}\text{C}^{-1}$.

Relevant theory

This constant is known as the linear expansivity of the material of the object and is defined as the fractional increase of length per unit rise of temperature.

Required equations

$\alpha = \frac{\Delta l}{l\,\Delta T}$

Applying it to this question

$\alpha = \dfrac{2.4\times10^{-5}}{100}$

Reasoning and calculation

$\alpha = \dfrac{2.4\times10^{-5}}{10^{2}}$

$\alpha = 2.4\times10^{-5-2}$

$2.4\times10^{-7}$

$\alpha$

Checks

$f / dT$

Check the unit.

$\Delta l/l$ has no unit.

Divide it by °C. Then $\alpha$ is in ${}^{\circ}\text{C}^{-1}$.

Does it make sense?

The numbers of the question give $\alpha = 2.4 \times 10^{-7}\ ^\circ\text{C}^{-1}$.

Common metals have $\alpha$ near $10^{-5}\ ^\circ\text{C}^{-1}$. So this rod expands very little.

But the answer must use the numbers given.

Physical plausibility

Conclusion

$\alpha = 2.4\times10^{-7}\ ^\circ\text{C}^{-1}$

$\alpha$ is the fractional change for $1\ ^\circ\text{C}$.

$$\alpha = \frac{2.4 \times 10^{-5}}{100} = 2.4 \times 10^{-7}\ ^\circ\text{C}^{-1}$$

The answer is option (5).

α = Δl/l

So divide by $\Delta\theta$.

$$\alpha = \frac{\Delta l/l}{\Delta\theta}$$

Explore

A metal rod is heated by $100\ ^\circ\text{C}$.

It gets a little longer. Its fractional change in length is $2.4 \times 10^{-5}$.

That is for $100\ ^\circ\text{C}$. For $1\ ^\circ\text{C}$ it is 100 times smaller.

So $\alpha = 2.4 \times 10^{-7}\ ^\circ\text{C}^{-1}$.