In the previous chapter, we described the motion of an object. We have not yet discovered what causes the motion. Why does the speed of an object change with time? Do all motions require a cause? If so, what is the nature of this cause? In this chapter we shall make an attempt to quench all such curiosities.
In our everyday life we observed that some efforts is required to put a stationary object into motion or to stop a moving object. We ordinarily experience this as a muscular effort and say that we must push or hit or pull on an object to change its state of motion. The concept of force is based on this push, hit or pull.
Let us now ponder about a "force". What is it? In fact, no one has seen, tasted or felt a force. However, we always see or feel the effect of a force. It can only be explained by describing what happens when a force is applied to an object.
A force can be used to change the magnitude of velocity of an object [that is, to make the object move faster or slower] or to change its direction of motion. We also know that a force can change the shape and size of objects.
For a lay man the term "mass" means the quantity of matter contained in a body. Newton gave a scientific meaning to the concepts of mass and force which are used in the formulation of the laws of motion. These principles of motion in three generalizations which are refered as Newtons Laws of Motion. These laws have formed the foundation of the science of Dynamics and Astronomy.
Dynamics is the study of causes of motion of bodies. The study of the motion of bodies moving at speeds that are very small compared to the speed of light is called classical mechanics. In such a study we apply Newton's laws of motion.
Newton's Laws of Motion
Newton's I Law
* According to I law, if no net force acts on the body, the acceleration of the body is zero i.e. the body may be in the state of rest or move with uniform velocity in the absence of external force.
* If F = 0; V = constant and a = 0
* In Newton's I law, there is no distinction between the absences of forces and the presence of forces whose resultant is zero.
* Newton I law gives the definition of force and inertia.
* Force: Force is that which changes (or) tends to change the state of rest (or) of uniform motion in a straight line.
Resultant force applied in a various states:
i) 
In this case only the magnitude of velocity of the particle changes whereas its direction remains same. Consequently the path of particle is a straight line.
ii) 
In this case only the direction of velocity of the particle changes, whereas its magnitude remains constant. Consequently the path of the particle is a circle.
iii) 
In this case both the magnitude as well as direction of velocity of the particle change. Consequently the path of motion of the particle is a helix.
* Unit of force: Newton in MKS System.
Dyne in CGS System
1 Newton = 105 dyne
* Dimensions of force: MLT-2
* Inertia: The inability of the body to change by itself from its state of rest or of uniform motion in a straight line is known as inertia.
* Mass is the measure of inertia i.e. greater the mass of the body, greater is the inertia of the body.
* Mass is the measure of inertia of a body in translational motion.
Moment of inertia is a measure of inertia of a rotating body.
* Examples: (i) When a stationary vehicle suddenly moves then the passenger inside the vehicle falls backward due to inertia of rest.
ii) If a moving vehicle suddenly stops then the passenger inside the vehicle bends forward due to inertia of motion.
iii) If a moving vehicle takes a turn passenger in, will give an outward jerk due to inertia of direction..
Momentum: To explain the action of a force on a body, Newton introduced the concept known as momentum, which he called "Quantity of motion".
i) It is the ability of the body to impart motion to another body.
ii) The momentum of a body is measured by the product of mass and velocity 
iii) It is a vector quantity.
iv) When a ball of mass 'm' hits a wall with a velocity 'v' and rebounds with the same velocity, then the change in momentum of the ball is equal to '2mv'.
This itself is the momentum imparted to the wall.
iv) When a ball of mass 'm' hits a wall with a velocity 'v' making an angle 'θ' with the wall and reflects with the same velocity making same angle with the wall, then the change in momentum of the ball is equal to '2mv sinθ' in the normally outward direction from wall.
Units of momentum: C.G.S. → gm. cm /sec (or) dyne. sec
M.K.S. → kg . m / sec (or) Newton. sec
F.P.S. → Pound . Ft / sec (or) Poundal. sec
Newton's II Law:
i) Newton's II law gives the magnitude of force.
ii) According to II law, "The rate of change of momentum of a body is directly proportional to the resultant or net external force acting on the body and takes place in the direction in which the force acts''.
iii) If two forces F1 and F2 act on a body making an angle θ with each other, then the resultant force is equal to
iv) If two forces F1 and F2 are acting at right angles to each other on a body of mass 'm', then the acceleration of the body is
v) Units of force: C.G.S. M.K.S.
Absolute unit → dyne Newton
Practical unit → gm.Wt kg. Wt
[g. dyne] [g.Newton]
vi) Force is a vector quantity.
vii) According to II law, F × t = P2 - P1 [change in momentum]
viii) If a car and lorry are initially moving with same momentum, then by the application of some breaking force, both will come to rest in same time.
ix) Two bodies, one is heavier and the other is lighter are moving with same momentum. If they are stopped by the same retarding force.
a) The distance travelled before coming to rest is
b) They travel different distances before coming to rest.
c) The heavier body covers lesser distance, the lighter body covers greater distance before coming to rest.
Newton's III Law
i) To every action force there is an equal and opposite reaction force.
Action = - Reaction
ii) Action and reaction are the forces acting on two different bodies. Hence they
don't cancel each other.
iii) Forces always exist in pairs.
iv) If action accelerates a body, then the reaction decelerates the other body.
v) Examples: a) Walking of a person on the road.
b) Pulling of a cart by a horse.
c) Jerk produced in a gun when a bullet is fired from it.
vi) Newton's III law leads to law of conservation of momentum.
Law of Conservation of Linear Momentum
i) If there are no external forces acting on the system, the total vector momentum of the system remains constant.
ii) If two bodies moving in same direction collide with each other then, total momentum before collision = Total momentum after collision m1u1 + m2u2 = m1v1 + m2v2
iii) If two bodies stick together after collision, then m1u1 + m2u2 = (m1 + m2) v
iv) If two bodies moving in opposite directions collide with each other and stick together after collision, thenm1u1 - m2u2 = (m1 + m2) v
* The time taken to come to rest, 
* The two bodies come to rest within same interval of time [ as P and F are same ]
Impulse
i) A large force acting for a very short interval of time is called impulse
ii) According to II law of motion, 
Thus, impulse is the product of the force and the time for which the force acts and is equal to the change in momentum of the body during that time.
iii) Example:
a) While catching a ball, a cricket player lowers his hands. If the total change in momentum is brought about in a very short interval of time, the average force is large.
.
But, by increasing the time interval, the average forceis decreased. It is for this reason that a cricketer lowers his hands while catching a ball and thus increases the time of catch so that he is not hurt.
iv) Impulse is equal to the integral of force with respect to time.
v) Area under force - time curve gives the magnitude of impulse.
vi) SI unit of impulse is newton. second or NS
vii) Impulse and momentum are vector quantities having same units and dimensions.
5. When a bullet is fired from a gun, then bullet and gun will have momenta equal in magnitude but opposite in direction.
m1v1 + m2v2 = 0
6) If a shell at rest explodes into two pieces, then two pieces will have momenta equal in magnitude but opposite in direction.
7) If a box sledge of mass M is travelling across the ice at a speed 'u' and a mass 'm' is dropped into it vertically, then the subsequent speed of the sledge is given by
8) Rocket and jet engine work on the principles of law of conservation of linear momentum.
Apparent weight of a person in a lift
The forces acting on the person standing in a lift are
i) His weight acting vertically downward.
ii) Reaction force (N) by the floor of the lift in the vertically upward direction.
The reaction force is the apparent weight of the body.
(a) When the lift is at rest (or) moving with uniform velocity:
The resultant force on the person = mg - N
As the lift is at rest the resultant force is zero
by Newton's I law.
0 = mg - N
Apparent weight N = mg
The apparent weight of a body in a lift moving with uniform velocity or at rest is equal to its real weight.
(b) When a lift going up with a uniform acceleration:
i) If the reactional force on the floor of the lift is N, then
N - mg = ma
ii) The apparent weight of the person in a lift going up with a uniform acceleration is more than its real weight.
iii) The resultant acceleration of the person in side the lift (g + a).
(c) When the lift is moving down with uniform acceleration:
(i) mg - N = ma
ii) The apparent weight of the person in a lift moving down with uniform acceleration is less than its real weight.
iii) A person feels lighter in a lift falling with uniform acceleration.
iv) The resultant acceleration of the person in side the lift is [g - a].
v) If the cable of the lift breaks then it moves down with acceleration g. In this condition the weight of the person is N = m[g - g] = 0. This is known as the state of weightlessness.
Connected Bodies
i) (a) If an object of mass M suspended by a very light string (we can neglect its mass) and acceleration of the object is zero, then the tension in the string is
(b) If the string is of mass 'm', the tension in the string
T = (M + m)g
ii) (a) A bucket of mass M pulled upward with an acceleration 'a' from a well by means of a rope of mass m, the tension in the string at the upper end
T = (M + m) (g + a)
This tension is equal to the force on our hands through which we pull the bucket up.
(b) If the bucket and rope system is allowed to move downward with an acceleration 'a', the tension at the upper end is
T = (M + m) (g - a)
iii) (a) When two bodies of masses m1 and m2 are attached at the ends of a string passing over a pulley as shown in the figure (Neglecting the mass of the pulley).
For m1:
T - m1g = m1a
For m2:
m2g - T = m2a
(b) If in the above system, mass (m) of the pulley is taken into account,
iv) If three blocks of masses m1, m2 and m3 are connected as shown in the figure.
for m1: T1 = m1a
for m2: T2 - T1 = m2a
for m3: T3 - T2 = m3a
T3 = (m1 + m2 + m3)a
(or) 
v) (a) Two objects connected by a string - One on a Horizontal surface and the other hanging vertically.
for m1: T = m1a
for m2: m2g - T = m2a
(b) 
Acceleration of the system 
vi. Blocks placed in contact with each other on frictionless Horizontal surface: Acceleration of the system 
Contact force between m1 and m2, F12
Acceleration of the system 
Contact force between m1 & m2, F12 = 
