Example 2

Remember that square brackets [ ] represent the units of.

\begin{aligned} \text{[f]}&=\text{[m][a]}\\&=\text{kg m s}^{-2}\end{aligned}

Example 3

Always make the physical quantity to be the subject of the formula.

\begin{aligned} k&=\frac{F}{x}\\\text{[k]} &= \frac{\text{kg m s}^{-2}}{\text{m}}\\&=\text{kg s}^{-2} \end{aligned}

Example 4(a)

To check whether the equation is balanced or not, work out the LHS and RHS separately. Most student just directly substitute the units, which makes the workings hard to follow.

\begin{aligned} LHS&=[p]\\&=\frac{[F]}{[A]}\\&=\frac{\text{kg m s}^{-2}}{\text{m}^{2}} \\&=\text{kg m}^{-1} \text{s}^{-2}\end{aligned}

\begin{aligned} RHS &=[d][\rho][g]\\&=\text{m}\text{ kg m}^{-3} \text{m s}^{-2}\\&=\text{kg m}^{-1}\text{s}^{-2}\end{aligned}

Since LHS = RHS, the equation is balanced.

Example 4(b)

To find the units of energy, just choose any energy equation that you know. Here, I am using $E=mgh$ :

\begin{aligned} \text{LHS} &= [m][g][h]\\&=\text{kg m s}^{-2} \text{m} \\ &=\text{kg m}^2 \text{s}^{-2} \end{aligned}

\begin{aligned} RHS &= \text{kg }\text{(m s)}^2\\&=\text{kg m}^2 \text{ s}^{-2} \end{aligned}

Homework Questions

Question 1

\begin{aligned} \text{LHS} &= \frac{\text{kg m s}^{-2}}{\text{m}^{2}}\\&=\text{kg m}^{-1}\text{s}^{-2} \end{aligned}

\begin{aligned} \text{RHS}&=\text{kg m}^{-3} \text{m s}^{-2} \text{m}\\&=\text{kg m}^{-1}\text{s}^{-2}\end{aligned}

Since LHS = RHS, the equation is balanced.

Question 2

Remember to always express the quantity as the subject of the formula.

\begin{aligned} v^2 &= \frac{\gamma p}{\rho}\\ \gamma &= \frac{v^2 \rho}{p}\end{aligned}

\begin{aligned} \gamma &= \frac{(\text{m s}^{-1})^2 \text{kg m}^{-3}}{\text{kg m}^{-1} \text{s}^{-2}} \\&=1\end{aligned}

Hence $\gamma$ has no units.