Table of contents  
Newton’s Law Of Universal Gravitation  
Properties of Gravitational Force  
Gravitation Equation Vector Form  
Mass and Weight  
Is the force of Gravity the same all over the Earth? 
In the 1600s, an English physicist and mathematician named Isaac Newton was sitting under an apple tree. Apparently, an apple fell on his head, and he started wondering why the apple was attracted to the ground in the first place. Today we all know that it happened due to the force of gravitation.
Newton and fallen Apple led to discovery of Gravity
Newton’s Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
The universal gravitation equation thus takes the form
Gravitation Equation
where F is the gravitational force between bodies, m_{1} is the mass of one of the objects, m_{2} is the mass of the second object, r is the distance between the centers of two objects, and G is the universal gravitational constant.
The constant proportionality (G) in the above equation is known as the universal gravitation constant. Henry Cavendish experimentally determined the precise value of G.
The value of G is found to be G = 6.673 x 10^{11} N m^{2}/kg^{2}.
The universal gravitation equation is represented in vector form as follows:
where G is the gravitational constant and r is the unit vector from m_{1} to m_{2}. The figure below shows the gravitational force on m_{1} due to m_{2} along with r where the vector r is (r_{2}–r_{1}).
If we have a collection of point masses, the force on any one of them is the vector sum of the gravitational forces exerted by the other point masses:
The gravitational force on point mass m_{1} is the vector sum of the gravitational forces exerted by m_{2}, m_{3} and m_{4}. Therefore, the total force on m_{1} is given by:
Example 1: Calculate the gravitational force of attraction between the Earth and a 70 kg man standing at sea level, a distance of 6.38 x 10^{6} m from the Earth’s center.
Solution:
Given,
m_{1} is the mass of the Earth = 5.98 x 10^{24} kg
m_{2} is the mass of the man = 70 kg
d = 6.38 x 10^{6} m
The value of G = 6.673 x 10^{11} N m^{2}/kg^{2}
Now substituting the values in the Gravitational force formula, we get
Example 2: A sphere of mass 100 kg is attracted by another spherical mass of 11.75 kg by a force of 19.6 x 10^{7} N when the distance between their centers is 0.2 m. Find G.
Solution: Mass of first body = m_{1} = 100 kg, mass of second body = m_{2} = 11.75 kg, distance between masses = r = 0.2 m, force between them = F = 19.6 x 10^{7} N,
To Find: Universal gravitation constant = G =?
By Newton’s law of gravitation
The value of the universal gravitation constant is 6.672 x 10^{11} N m^{2}/kg^{2}.
Example 3: Calculate the gravitational force of attraction between two metal spheres each of mass 90 kg, if the distance between their centers is 40 cm. Given G = 6.67 x 10^{11} N m^{2}/kg^{2}. Will the force of attraction be different if the same bodies are taken on the moon, their separation remaining the same?
Solution: Mass of first body = m_{1} = 90 kg, mass of second body = m_{2} = 90 kg, Distance between masses = r = 40 cm = 40 x 10^{2} m, G = 6.67 x 1011 N m^{2}/kg^{2} .
To Find: Force of attraction = F =?
By Newton’s law of gravitation
If the same bodies are taken on the moon, their separation remains the same, the force of attraction between the two bodies will remain the same because the force of attraction between two bodies is unaffected by the presence of the third body and medium between the two bodies.
The force of attraction between two metal spheres is 3.377 x 10^{6} N.
Example 4: Three 5 kg masses are kept at the vertices of an equilateral triangle each of side of 0.25 m. Find the resultant gravitational force on any one mass. G = 6.67 x 10^{11} S.I. units.
Solution: m_{1} = 5 kg, m_{2} = 5 kg, m_{3} = 5 kg, r = 0.25 m, G = 6.67 x 10^{11} N m^{2}/kg^{2}.
To find: Force on m1 =?
By Newton’s law of gravitation, the force on mass m_{1} due to mass m_{2}.
By Newton’s law of gravitation, the force on mass m_{1} due to mass m_{3}.
The angle between F_{12} and F_{13} is 60°. (Angle of an equilateral triangle). The net force on m_{1} is given by
The two forces are equal, hence their resultant act along angle bisector towards centroid.
Force on any mass is 4.621 x 10^{8} N towards the centroid
Example 5: Two bodies of masses 5 kg and 6 x 10^{24} kg are placed with their centres 6.4 x 10^{6} m apart. Calculate the gravitational force of attraction between the two masses. Also, find the initial acceleration of two masses assuming no other forces act on them.
Solution: Mass of first body = m_{1} = 5 kg, mass of second body = m_{2} = 6 x 10^{24} kg, Distance between masses = r = 6.4 x 10^{6} m, G = 6.67 x 10^{11} N m^{2}/kg^{2} .
To Find: Force of attraction between two masses = F =? Initial accelerations of the two masses =?
Ans:
By Newton’s law of gravitation
Initial acceleration of the body of mass 5 kg
By Newton’s second law of motion F = ma
Thus a = F/m = 48.85 / 5 = 9.77 m/s^{2}
Initial acceleration of the body of mass 6 x 10^{24} kg
By Newton’s second law of motion F = ma
Thus a = F/m = 48.85 / 6 x 10^{24} = 8.142 x 10^{24} m/s^{2}
The force of attraction between the two masses = 48.85 N
In Newton’s law of gravity, we noticed that the mass is a crucial quantity. We consider mass and weight to be the same, but they are related but are different in reality.
Difference between Mass and Weight
The forces of speed and gravity are what keeps the moon in constant orbit around the Earth. The Moon seems to hover around in the sky, unaffected by gravity. However, the reason the Moon stays in orbit is precisely because of gravity. In this video, clearly, understand why the moon doesn’t fall into the earth
Gravity isn’t the same everywhere on earth. Gravity is slightly stronger over the places with more underground mass than places with less mass. NASA uses two spacecraft to measure the variation in the Earth’s gravity. These spacecraft are a part of the Gravity Recovery and Climate Experiment (GRACE) mission.
Areas in blue have weaker gravity while areas in red have slightly stronger gravity.
118 videos470 docs189 tests

1. What is Newton's Law of Universal Gravitation? 
2. What are the properties of gravitational force? 
3. What is the gravitational equation in vector form? 
4. What is the difference between mass and weight? 
5. Is the force of gravity the same all over the Earth? 
118 videos470 docs189 tests


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