Question

An equation for the period of a planet is 4 pie² r³/Gm where T is in secs, r is in meters, G is in m³/kgs² m is in kg, show that the equation is dimensionally correct.​

Answer 1

Answer:

$\displaystyle T = \sqrt{\frac{4\, \pi^{2} \, r^{3}}{G \cdot m}}$.

The unit of both sides of this equation are $\rm s$.

Explanation:

The unit of the left-hand side is $\rm s$, same as the unit of $T$.

The following makes use of the fact that for any non-zero value $x$, the power $x^{-1}$ is equivalent to $\displaystyle \frac{1}{x}$.

On the right-hand side of this equation:

• $\pi$ has no unit.
• The unit of $r$ is $\rm m$.
• The unit of $G$ is $\displaystyle \rm \frac{m^{3}}{kg \cdot s^{2}}$, which is equivalent to $\rm m^{3} \cdot kg^{-1} \cdot s^{-2}$.
• The unit of $m$ is $\rm kg$.

$\begin{aligned}& \rm \sqrt{\frac{(m)^{3}}{(m^{3} \cdot kg^{-1} \cdot s^{-2}) \cdot (kg)}} \\ &= \rm \sqrt{\frac{m^{3}}{m^{3} \cdot s^{-2}}} = \sqrt{s^{2}} = s\end{aligned}$.

Hence, the unit on the right-hand side of this equation is also $\rm s$.