Question

newtons law of universal gravitation is represented by f = gmm/r² where f is the magnitude of the gravitational force exerted by one small object on another, m and m are the masses of the objects, and r is a distance. force has the si units kg·m/s². what are the si units of the proportionality constant g?

Explanation:

Step1: Recall Newton's law of universal gravitation formula and unit - analysis

The formula is $F=\frac{GMm}{r^{2}}$, and the SI unit of force $F$ is $\text{kg}\cdot\text{m/s}^{2}$, the SI units of mass $M$ and $m$ are $\text{kg}$, and the SI unit of distance $r$ is $\text{m}$.

Step2: Rearrange the formula for $G$

We can rewrite $F = \frac{GMm}{r^{2}}$ as $G=\frac{Fr^{2}}{Mm}$.

Step3: Substitute the SI - units of each quantity into the formula for $G$

Substitute the units: $F$ has units $\text{kg}\cdot\text{m/s}^{2}$, $r$ has units $\text{m}$, $M$ and $m$ have units $\text{kg}$. Then $G$ has units $\frac{(\text{kg}\cdot\text{m/s}^{2})\cdot\text{m}^{2}}{\text{kg}\cdot\text{kg}}$.

Step4: Simplify the unit expression

$\frac{(\text{kg}\cdot\text{m/s}^{2})\cdot\text{m}^{2}}{\text{kg}\cdot\text{kg}}=\frac{\text{kg}\cdot\text{m}^{3}}{\text{kg}^{2}\cdot\text{s}^{2}}=\text{m}^{3}/(\text{kg}\cdot\text{s}^{2})$.

Answer:

$\text{m}^{3}/(\text{kg}\cdot\text{s}^{2})$