What is gravitational force?

In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.

Gravitational force can be described using newton’s universal law of gravitation.

Newton’s law of gravitational force

Let us consider two masses $m_1 and m_2$  that is kept at a distance r away from each other. Let F be the magnitude of the gravitational force of attraction.                                                                     

Newton's law of gravitation states that every object in the universe attracts every other object with a force F. This force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them.

Let F be the magnitude of the force.

i. e,    $\mathrm{F\alpha } m_1{m}_2$                   (1)                        

Magnitude of the force is inversely proportional to the square of the distance between the center of the two masses.

i.e, $F\alpha \frac{1}{r^2}$                       (2)

Now, equating the equation (1) & (2),

$F\alpha \frac{m_1{m}_2}{r^2}$

$F=G\frac{m_1{m}_2}{r^2}$

where,

G is the gravitational constant.

Value of the gravitational constant (G) is $6.67\times {10}^{-11} N{m}^2 k{g}^{-2}.$

Vector form of newton’s gravitational force

Consider two bodies of masses $m_1$ and $m_2$. Let $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ be the position vectors of the two bodies.

From the figure,

${\overrightarrow{r}}_{21}=\left|{\overrightarrow{r}}_2-{\overrightarrow{r}}_1\right|$

$\overrightarrow{r_{21}}$ is the vector joining the two bodies.

The force exerted by the object of mass m₁ on mass m₂ is,

${\overrightarrow{F}}_{21}=-G\frac{m_1{m}_2}{{r_{21}}^2}{\hat{r}}_{21}$                                       

The negative sign indicates that the force is attractive.

Similarly, the force exerted by the mass m₂ on mass m₁ is,

${\overrightarrow{F}}_{12}=-G\frac{m_1{m}_2}{r_{12}^2}{\hat{r}}_{12}$

Since, ${\widehat{r }}_{12}=-{\hat{r}}_{21}$

  $$\begin{gathered} {\overrightarrow{F}}_{12}=-G\frac{m_1{m}_2}{\left(r_{12}^2\right)}\left(- {\hat{r}}_{21}\right) \\ {\overrightarrow{F}}_{12}=G\frac{m_1{m}_2}{r_{21}^2} {\hat{r}}_{21} \\ {\overrightarrow{F}}_{12}=-{\overrightarrow{F}}_{21} \end{gathered}$$

Hence, the mutual forces exerted by the objects are equal in magnitude and opposite in direction. 

So, the newton’s gravitational force obeys newton's third law of motion.

Gravitational force formula

According to Newton’s law of gravitation, every object in the universe attracts each other with a specific force that is directly proportional to the product of the two masses and is inversely proportional to the square of the distance between the two masses.

It is expressed by a mathematical expression,

$F=G\frac{m_1{m}_2}{r^2}$

Where,

F is the magnitude of the gravitational force (Newton or N).

G is the gravitational constant.

m₁ is the mass of the first object (kg).

m₂ is the mass of the second object (kg).

r is the distance between the two masses (m).

Acceleration due to gravity

When we throw an object upwards, we observe that the object reaches a certain height and falls back. The object falls back due to gravity. This type of motion is called free-fall motion. The object gets accelerated due to gravity. This acceleration is independent of the mass of the object. The acceleration due to gravity is indicated by g and the S.I unit is m/s².

Acceleration due to gravity is a vector quantity. It is always directed towards the center of the earth. The acceleration due to gravity is independent of the mass of the body. The value of gravity changes with the altitude i.e. the height from the surface. The value of 'g' is taken as$9.8 \mathrm{m}/{\mathrm{s}}^2$ for all the experimental purposes. 

The acceleration due to gravity is different for different planets. The acceleration due to the gravity of the moon is one-sixth of the Earth and 27 times that of the Sun. Comparatively, the mercury has minimum accelerations due to the gravity in the entire planets of the solar system.

Now, the relation between acceleration due to gravity and the gravitational constant is given by,

$g=\frac{GM}{R^2}$

Where,

M is the mass of the earth.

R is the radius of the earth.

Gravitational Field

The gravitational field is a vector field, every point around an object is associated with a vector directed towards the object. It is defined as the force exerted by an object due to an object with unit mass.

Intensity of gravitational field

The strength of a gravitational field acting per unit mass or masses is called the intensity of the gravitational field.

The intensity of the gravitational field is denoted by E_g. It is expressed by,

$E_g=\frac{F}{m}$

Gravitational potential

The amount of work done per unit test mass from infinity into the gravity of source mass or mass is called gravitational potential.

The gravitational potential is denoted by V_g and it is expressed as,

$V_g=-\frac{GM}{r}$

Here, the gravitational potential is always negative.

Gravitational potential Energy

The gravitational potential energy or gravitational energy is the energy possessed by an object due to its position with respect to the gravitational field. When a body of mass m is moved from infinity to a point under the influence of gravity, without accelerating it, then the amount of work done in moving the object is stored as potential energy.

The gravitational potential energy is denoted by U. It is given by,

$U=mgh$

where,

m is the mass of the object (kg).

g is the acceleration due to gravity (on the surface of the earth, g=9.8m/s²).

 h is the height of the object (m).

Formulas

The gravitational force is represented as,

$F=\frac{G{m}_1{m}_2}{r^2}$

The relation between gravity and acceleration is given by,

$g=\frac{GM}{R^2}$

The intensity of the gravitational field is expressed by,

$E_g=\frac{F}{m}$

The gravitational potential is expressed by,

$V_g=-\frac{\mathrm{GM}}{r}$

Context and Applications

This topic is one of the important applications in all forms of classical mechanics and it is significant for both undergraduate and postgraduate courses, especially for:

Bachelors in Science (Physics)

Masters in Science (Physics)

Practice Problems

Question 1: What is the force of gravity implemented on an object of mass 1000 kg at the earth's surface? Here, mass of the earth is $5.98\times {10}^{24}$ kg and radius is $6.38\times {10}^6$ m?

1. 9.817 N
2. 8.92 N
3. 5.61 N
4. 10.26 N

Given data:

The mass of the earth (m₁) =$5.98\times {10}^{24}$ kg

The mass of the object (m₂) =$1000$ kg

The radius of the earth (r) =$6.38\times {10}^{6}m$

Acceleration due to gravity=$9.8 m/s^2$

Universal gravitational constant (G) = $6.67\times {10}^{-11}N{m}^2/k{g}^2$

Solution:

We know that,

$$\begin{gathered} F=\frac{G{m}_1{m}_2}{r^2} \\ F=\frac{6.67\times {10}^{-11}\times 5.98\times {10}^{24}\times 1000}{{\left(6.38\times {10}^{24}\right)}^2} \\ F=9.817N \end{gathered}$$

Answer: The correct option is (a).

The gravitational force is 9.81 N.

Question 2: The newton’s gravitational law applies to ________ .

1. Planet only
2. Small bodies only
3. For solar system
4. Both small and big bodies

Answer: The correct option is (b).

Explanation: Newton's gravitational law states that any two objects in the universe attracts each other with a specific set of forces that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Since the objects maybe smaller or bigger, it is applicable for both small and big objects.

Question 3: The value of gravitational constant is _______ .

1. $9.8 m/s^2$
2. $6.371\times {10}^6 m$
3. $6.673\times {10}^{-11}N{m}^{2}k{g}^{-2}$
4. $1.564\times {10}^{-11} N{m}^{2}k{g}^{-2}$

Answer: The correct option is (c).

Explanation:  The universal gravitational constant (G) is equal to $6.673\times {10}^{-11} N{m}^2 k{g}^{-2}$.

Question 4: The gravitational force F is inversely proportional to ________.

1. Product of their mass
2. Square of the distance between the masses
3. Product of the distance
4. Sum of the distance and mass

Answer: The correct option is b.

Explanation: The gravitational force F is directly proportional to the product of the masses and is inversely proportional to the square of the distance between them.

Question 5: Acceleration due to gravity decreases with increase in_________.

1. Altitude
2. Depth
3. Amplitude
4. Mass

Answer: The correct option is a.

Explanation: Acceleration due to gravity decreases with an increase in the altitude above the earth's surface.