This law states that the attractive force between two bodies is directly proportional to the product of the their two masses ($m_1\times m_2$) and inversely proportional to the square of their distance apart ($r^2$).
This law can be written as a proportionality: $$ \begin{equation}\begin{aligned} F\propto \frac{m_1m_2}{r^2}\\ \end{aligned}\end{equation} $$ Introducing a constant of proportionality, $G$, known as the gravitational constant, we get: $$ \begin{equation}\begin{aligned} F_g= \frac{Gm_1m_2}{r^2}\\ \end{aligned}\end{equation} $$
The value of $G$ is $6.67\times 10^{-11} Nm^2kg^{-2}$
Gravitational field strength
Consider an object of mass, $m$ being pulled by a planet or other celestial body with mass, $m$ separated by a distance of $r$: $$ \begin{equation}\begin{aligned} F_g= \frac{GMm}{r^2}\\ \end{aligned}\end{equation} $$
The gravitational force is the weight of this object: $$ \begin{equation}\begin{aligned} F_g=mg\\ \end{aligned}\end{equation} $$
The gravitational field strength, a.k.a. acceleration due to gravity is the force per unit mass: $$ \begin{equation}\begin{aligned} mg&=\frac{GMm}{r^2}\\ g&=\frac{GM}{r^2}\\ \end{aligned}\end{equation} $$
Note how the gravitational field strength is independent of (not affected by) the mass of the object - the mass of the planet and the separation distance are the only two variables involved.
Gravity as centripetal force
The gravitational force, $F_g$ can be thought of as the centripetal force ($\frac{mv^2}{r}$) responsible for keeping objects orbiting around the Earth (or other heavenly body) in their circle of revolution:
$$ \begin{equation}\begin{aligned} F_g&=F_c\\ \frac{GMm}{r^2}&=\frac{mv^2}{r}\\ \end{aligned}\end{equation} $$If we cancel the object’s mass, $m$ from both sides of the equation and make velocity the subject of the formula, we get: $$ \begin{equation}\begin{aligned} \frac{GM}{r^2}&=\frac{v^2}{r}\\ v^2&=\frac{GM}{r}\\ v&=\sqrt{\frac{GM}{r}}\\ \end{aligned}\end{equation} $$
What do you think?
1) If $v=\omega r$ and $v=\sqrt{\frac{GM}{r}}$, what is $\omega$?
- $\omega=\sqrt{\frac{GM}{r^3}}$
- $\omega=\sqrt{\frac{GM}{r^2}}$
- $\omega=\sqrt{\frac{GMr}{r}}$
- $\omega=\sqrt{\frac{GM}{r^2}}r$