Explanation: When a body is at rests on the surface of the Earth, it is acted upon by the gravitational force of the Earth. Let us compute the magnitude of this force in two ways. Let 'M' be the mass of the Earth and 'm' be the mass of the body. The entire mass of the Earth is assumed to be concentrated at its centre. The radius of the Earth is 6378 km (=6400 km approximately). By Newton's law of gravitation, the force acting on the body is given by

\(F=\frac{G M m}{R^{2}}\)               ................(1)
Here, the radius of the body considered is negligible when compared with the Earth's radius. Now, the same force can be obtained from Newton's second law of motion. According to this law, the force action on the body is given by the product of its mass and acceleration (called as weight). Here acceleration of the body is under the action of gravity, hence a = g.
F = Ma = mg.
F = weight = mg       ................(2)
comparing (1) and (2) we get,
ma = GMm
\(\mathrm{mg}=\frac{\mathrm{GMm}}{\mathrm{R}^{2}}\)
Acceleration due to gravity,
\(g=\frac{G M}{R^{2}}\)

Newton's third law states that "for every action, there is an equal and opposite reaction". They always act on two different bodies.
If a body 'A' applies a force F_A on a body 'B', then the body 'B' reacts with force F_B on the body ‘A’, which is equal to F_A in magnitude but opposite in direction. F_B  = -F_A.
Examples :
(i) When birds fly, they push the air downwards with their wings (Action) and the air pushes the bird upward (Reaction).
(ii) When a person swims, he pushes the water using the hands backwards (Action) and the water pushes the swimmer in the forward direction (Reaction).
(iii) When you fire a bullet, the gun recoils backward. The bullet is moving forward (Action) and the gun equalises this forward action by moving backward (Reaction).

A large force acting for a very short interval of time is called as Impulsive force.
When a force 'F' acts on a body for a period of time 't' then the product of force and time is Known as ‘impulse’ represented by 'J'.
Impulse J = F x t        ..............(1)
By Newton's second law
F = \(\triangle\)P/t (\(\triangle\) refers to change)
\(\triangle\)P = F x t                 ...............(2)
From 1 & 2, J = P\(\triangle\)
Impulse is also equal to the magnitude of change in momentum. It's unit is Kgms^-1 (or) Ns. Change in momentum can be achieved in 2 ways. They are;
(i) A large force acting for a short period time and
(ii) A smaller force acting for a longer period of time.
Examples :
(i) Automobiles are fitted with springs and shock absorbs to reduce jerks while moving on uneven roads.
(ii) In cricket, a fielder pull back his hands while catching the ball. He experiences a smaller force for a longer interval of time to catch the ball, resulting in a lesser impulse on his hands.

(i) At equilibrium, the algebraic sum of the moments of all individual forces about any point is equal to zero.

(ii) In the illustration, the force F₁, produces an anticlockwise rotation at a distance d, from the point of pivot (P) called fulcrum and force F₂ produces a clockwise rotation at a distance d₂ from the point of pivot P. The principle of moments can be written as follows;
Moment in Moment in
Clockwise direction = Anticlockwise direction
F₁ x d₁ = F₁ x d₂

i) Gears:
A gear is a circular wheel with teeth around its rim. It helps to change the speed of rotation of a wheel by changing the torque and helps to transmit power.
ii) Seesaw:
In Seesaw, there is a difference in the weight of the persons sitting on it, the heavier person lifts the lighter person. When the heavier person comes closer to the pivot point the distance of the line of action of the force decreases. It causes less amount of torque to act on it. This enables the lighter person to lift the heavier person.
iii) Steering Wheel:
A small steering wheel enables you to manoeuore a car easily by transferring a torque to the wheels with less effort.

Initial mass of the rocket, m = 20,000 kg
Initial acceleration, a = 50 ms^-2
(Upward direction)
Let initial thrust of the blast be T
To find: T = mg + ma
T = m(g + a) = 20,000 (9.8 + 50)
T = 2 x 10⁴ x 14.8
T = 29.6 x 10⁴ N

A shooting star is a small piece of rock that hits earth's atmosphere. It heats up due to air temperature. They enter with very high speeds. When it strikes with high speed in short duration (i.e. impulse = p = λt) causes burning. But when hit the ground, it becomes cool.

If a body moves with uniform velocity, the acceleration of body is zero.
∴ net force acting on the body is zero.
F = ma [a = 0]

Recoil velocity of a gun α\(\frac { 1 }{ m } \). So light rifle recoils with large velocity than the heavy rifle.

Application of Newton's law of gravitation
(i) Dimensions of the heavenly objects can be measured using gravitation law. Mass of the earth, radius of the earth, acceleration due to gravity etc. can be calculated with a higher accuracy.
(ii) Helps in discovering new stars and planets. Mass of the double stars can be calculated.
(iii) One of the irregularities in the motion of stars is called "Wobble" which leads to the disturbance in the motion of planet nearby. In this condition mass of the star can be calculated using law of gravitation.
(iv) Helps to explain germination of roots due to the property of geotropism, which is the property of root responding to the gravity.
(v) Helps to predict the path of the astronomical bodies.