2018 A/L Physics Past Paper · Question 02 (English Medium)

Homogeneity of an equation — dimensions

$X, Y$ and $Z$ represent three physical quantities with different dimensions. They can be combined to form another physical quantity $P$ of the form,

$P = AX + BY + CZ$

Which of the following expressions has different dimensions from the rest?

Answer and explanation

Answer: (5) $(BY)(CZ)$

What the question asks

Four of the five expressions have the same dimensions.

One is different. Find it.

Symbolic expressions which show the relation of a physical quantity to basic units is called its dimension.

If the equation is correct, the dimension of its both sides should be equal.

$P = AX + BY + CZ$ is a correct equation.

You can only add things that have the same dimensions. So every term has the dimensions of $P$ .

$[AX] = [BY] = [CZ] = [P]$

Every term is $[P]$ . So each expression is a power of $[P]$ .

asked

[P]

$AX = BY = CZ$

$AX$

$AX - CZ$

[P][P]/[P] = [P]

$(AX)(CZ)/(BY)$

[P]^2/[P] = [P]

$(BY)^2/P$

[P]^2

$(BY)(CZ)$

Options (1) to (4) all come out as $[P]$ .

Only option (5) is $[P]^{2}$ .

Check (5).

$[(BY)(CZ)] = [P] \times [P] = [P]^{2}$

$P$ has dimensions. So $[P]^{2}$ is not $[P]$ .

Check with a unit. Say $P$ is an energy in J.

Then (1) to (4) are in J. But (5) is in $J^{2}$ .

$[(BY)(CZ)] = [P]^2$

$(BY)(CZ)$ has dimensions $[P]^{2}$ .

The other four have $[P]$ . The answer is (5).

Adding and taking away do not change dimensions.

Only multiplying and dividing change them.

Exam tip