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### Physics For Scientists & Engineers

Book edition 9th Edition
Author(s) Raymond A. Serway, John W. Jewett
Pages 1624 pages
ISBN 9781133947271

# Newton’s law of universal gravitation is represented by ${F}{=}\frac{GMm}{{r}^{2}}$whereF is the magnitude of the gravitational force exerted by one small object on another, ${M}$ and are the masses of the objects, and ${r}$ is a distance. Force has the SI units ${\text{Kg}}{\cdot }{{\text{m/s}}}^{{\text{2}}}$ . What are the SI units of the proportionality constant role="math" localid="1663610883447" ${G}$?

The SI unit of the gravitational constant $G$ is:

$G=\frac{{\text{m}}^{\text{3}}}{\text{kg}\cdot {\text{s}}^{\text{2}}}$

See the step by step solution

## Step 1: Analysis of given data

The formula of the Newton’s law of gravitation is $F=\frac{GMm}{{r}^{2}},$ where $F$ is the gravitational force, $M$ and $m$ are the masses of bodies, and $r$ is the distance between the bodies. ………..(1)

It has given that the dimension of force is $\text{kg}\cdot {\text{m/s}}^{\text{2}}.$

The analysis of a relationship between different physical quantities by using the units of measurements and dimensions is called dimensional analysis. It is used to examine the correctness of an equation.

## Step 2: Write the units of physical quantity

The unit of mass is $\text{kg}$

The unit of length is $\text{m}$

The unit of time is $\text{s}$

## Step 3: Determination of units of gravitational constant  G

Rewrite the equation (1).

$F=\frac{GMm}{{r}^{2}}$

Write the SI units for the quantities in the above formula.

$\frac{\text{kg}\cdot \text{m}}{{\text{s}}^{\text{2}}}\text{=}\frac{G\cdot \text{kg}\cdot \text{kg}}{{\text{m}}^{\text{2}}}$

Rearrange the above equation for $G$ and cancel the equal terms.

$G=\frac{{\text{m}}^{\text{3}}}{\text{kg}\cdot {\text{s}}^{\text{2}}}$

Hence, the SI unit of the gravitational constant $G$ is:

$G=\frac{{\text{m}}^{\text{3}}}{\text{kg}\cdot {\text{s}}^{\text{2}}}.$