Newton's Law Of Gravitation - Practice Questions & MCQ

• According to Newton's law of gravitation, the gravitational force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Due to gravitational force, Each particle in this universe attracts every other particle.

The direction of this force is along the line joining the particles. 

Let two particles of masses $m_1$ and $m_2$ separated by a distance r exert a Force F on each other
And Magnitude of F is given as

$
\begin{array}{r}
F \propto \frac{m_1 m_2}{r^2} \\
\text { Or, } F=\frac{G m_1 m_2}{r^2}
\end{array}
$

Where
$F \rightarrow$ Force
$G \rightarrow$ Gravitational constant
$m_1, m_2 \rightarrow$ Masses
$r \rightarrow$ Distance between masses

•  The vector form of formula

According to Newton's law of gravitation

$
\begin{aligned}
& \qquad \vec{F}_{12}=\frac{-G m_1 m_2}{r^2}\left(\hat{r}_{21}\right) \\
& \vec{F}_{12}=\frac{-G m_1 m_2}{r^3}\left(\vec{r}_{21}\right) \\
& \text { Or } \\
& \text { Where } \hat{r}_{21} \rightarrow \text { Position vector }
\end{aligned}
$

Here negative sign indicates that the direction of $\vec{F}_{12}$ is opposite to that of $\vec{r}_{21}$
And $\hat{r}_{12}=-\hat{r}_{21}$
So, $\vec{F}_{12}=-\vec{F}_{21}$
Means Gravitational force between two bodies form an action and reaction pair.
i.e. the forces are equal in magnitude but opposite in direction.

This is in accordance with Newton's third law of motion
- Universal Gravitational Constant (G)

If $m_1=m_2=1 \mathrm{~kg}$ and $\mathrm{r}=1 \mathrm{~m}$ then $\mathrm{F}=\mathrm{G}$
I.e Universal gravitational constant is equal to the Gravitational force between two bodies each having unit mass and their centers are placed unit distance apart.

Value of G is $6.67 \times 10^{-11} \mathrm{~N}-\mathrm{m}^2 \mathrm{~kg}^{-2} \quad$ (S.I.)
Its Dimension Formula is $\left[M^{-1} L^3 T^{-2}\right]$

Value of G does not depend upon the nature & size of bodies

        Also, it is also independent of the nature of medium between two bodies.

• Properties of Gravitational Force

1. Always attractive

2. It is the central force

3.  Weakest force (Ratio of $F_g$ to $F_e$ between two electrons is $10^{-43}$ )
   4. It is a conservative force
   5. It is independent of the medium between the particles.
   6. Gravitational force is long range-force.
   7. Principle of superposition is valid for Gravitational Force

   The gravitational force between two particles is independent of the presence or absence of other particles.
   - $\mathrm{m}_1$
   - $\mathrm{m}_2$
   - $\mathrm{m}_3$

   Force on a particle $(\mathrm{m})$ due to the number of particles ( $m_{1,} m_2 m_3$, etc)
   is the resultant of forces due to individual particles ( $F_1, F_2, F_3$, etc)
   l.e- $\vec{F}=\vec{F}_1+\vec{F}_2+\vec{F}_3$ $\qquad$