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

`\displaystyle T = \sqrt{(4\, \pi^(2) \, r^(3))/(G \cdot m)}`.

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

Step-by-step 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 (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 (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{((m)^(3))/((m^(3) \cdot kg^(-1) \cdot s^(-2)) \cdot (kg))} \\ &= \rm \sqrt{(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`.