Gravitation - Notes, Topics, Formulas, Books, FAQs

Have you ever wondered how you can walk on the ground freely? or when a fruit falls from a tree or something drops from a height, it always heads for the ground? This is exactly what Sir Isaac Newton wondered and offered the world the concept of gravitation. In simple words, Gravitation is the force operating between two objects on the earth. So any thrown object will be attracted toward the earth and will fall on earth. The gravitational force has a very important role which allows us to live in this wonderful world. If gravity disappeared suddenly then we would all start to float immediately. So it is very important to study Gravitational.

Newton's Law of Gravitation

Gravitational force is an attractive force acting between two masses $m_1$ and $m_2$ separated by a distance $r$.

Directly proportional to the product of their masses, i.e., $F \approx\left(m_1 m_2\right) \ldots$ (1)
Inversely proportional to the square of the distance between their centre, i.e., $\left(F=1 / r^2\right) \ldots$. (2)

On combining equations (1) and (2), we get,

\begin{aligned}
& F \propto \frac{m_1 m_2}{r^2} \ldots(1) \\
& F=G \times \frac{m_1 m_2}{r^2} \ldots (2)
\end{aligned}
Then the Gravitational force $F=\frac{G m_1 m_2}{r^2}$

After studying that Gravitation force operates between any two objects on the earth, You may wonder and ask why not all bodies attract each other and stick together due to gravitational force. So answer to this question is that Gravitational Force is the weakest among all the four fundamental forces known to us now.

Let's Suppose 2 similar balls of mass 1 kg are placed 1m apart then the gravitational force acting between them is simply G which is equal to $6.6710 * 10^{-11} \mathrm{Nm}^2(\mathrm{~kg})^{-2}$ As you see this is a very negligible attractive force.

Vector Form of Newton's Law of Gravitation

The vector form of Newton's law of gravitation signifies that the gravitational forces acting between the two particles form an action-reaction pair.

From the above figure, it can be seen that the two particles of masses are placed at a distance.

$
\overrightarrow{r_{21}}=\left(\overrightarrow{r_2}-\overrightarrow{r_1}\right)
$
The direction of the vector is from $m_1$ to $m_2$
Therefore, the force applied on $\mathrm{m}_2$ by $\mathrm{m}_1$ is

$
\overrightarrow{F_{21}}=-\frac{G m_1 m_2}{r_{21}^2} r_{21}
$
The negative sign indicates the attractive nature of the force.
Similarly, force on $\mathrm{m}_1$ and $\mathrm{m}_2$

$
\vec{F}_{12}=-\frac{G m_1 m_2}{r_{12}^2} r_{\hat{12}}
$
Since,

$
\hat{r_{12}}=-\hat{r_{21}}
$
$
\vec{F}_{12}=-\frac{G m_1 m_2}{\left(-r_{21}\right)^2}\left[-\hat{r}_{21}\right]
$
$
\begin{aligned}
& \vec{F}_{12}=\frac{G m_1 m_2}{\left(r_{21}\right)^2}\left[r_{21}\right] \\
& =-\overrightarrow{F_{21}}
\end{aligned}
$
Hence, the applied forces are equal and opposite. Also, the gravitational force follows Newton's third law.

• Let's have a look at how g value changes due to the shape of the earth

$R_e \rightarrow$ Radius of equator
$R_p \rightarrow$ Radius of pole

$
R_{\text {equator }}>R_{\text {pole }}
$
The Equatorial radius is about 21 km longer than the polar radius.
So $g_{\text {pole }}>g_{\text {equator }}$
In fact

$
g_p=g_e+0.018 \mathrm{~m} / \mathrm{s}^2
$

Or we can say that Weight increases as the body is taken from equator to pole.

You will also learn about satellites and their motion in space. You will also learn to calculate their orbital velocity, their escape velocity, etc.

Related topics,

Summary

According to Sir Isaac Newton, gravity is that force which pulls two masses. The law of gravitation as given by Newton states that: this force is directly proportional to the product of mass and inversely proportional to distance squared. Gravitation is weak but universal; it affects all objects in the universe equally regardless of their size or composition. This force obeys Newton’s third law of motion, i.e., for every action, there is an equal but opposite reaction; it also varies with different shapes of the earth such that it is stronger at the poles than at the equator.

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