Newton's Laws of motion 177

Chapter

4
Newton's Laws of Motion
Point Mass If a body of mass m is moving with velocity v

(1) An object can be considered as a point object if then its linear momentum p is given by p mv

during motion in a given time, it covers distance much (4) It is a vector quantity and its direction is the
greater than its own size. same as the direction of velocity of the body.
(2) Object with zero dimension considered as a (5) Units : kg-m/sec [S.I.], g-cm/sec [C.G.S.]
point mass.
1
(6) Dimension : [MLT ]
(3) Point mass is a mathematical concept to
simplify the problems. (7) If two objects of different masses have same
momentum, the lighter body possesses greater velocity.
Inertia
p m1v1 m2v2 = constant
(1) Inherent property of all the bodies by virtue of
which they cannot change their state of rest or uniform v1 m2
motion along a straight line by their own is called
v2 m1 v
inertia.
(2) Inertia is not a physical quantity, it is only a 1
i.e. v p = constant
property of the body which depends on mass of the m
body. [As p is
(3) Inertia has no units and no dimensions constant]
(4) Two bodies of equal mass, one in motion and (8) For a given body
m
another is at rest, possess same inertia because it is a p v Fig : 4.1

factor of mass only and does not depend upon the (9) For different bodies moving with same
velocity. velocities p m
Linear Momentum p p
m= v=
(1) Linear momentum of a body is the quantity of
constant constant
motion contained in the body.
(2) It is measured in terms of the force required to
stop the body in unit time.
(3) It is also measured as the product of the mass
of the body and its velocity i.e., Momentum = mass Fig : 4.2 v Fig : 4.3 m

velocity.
Newtons First Law
 178 Newton's Laws of Motion
A body continue to be in its state of rest or of (a) If the string B is pulled with a sudden jerk then
uniform motion along a straight line, unless it is acted it will experience tension while due to
upon by some external force to change the state. inertia of rest of mass M this force will not A
(1) If no net force acts on a body, then the velocity be transmitted to the string A and so the
of the body cannot change i.e. the body cannot string B will break. M
accelerate. (b) If the string B is pulled steadily
B
(2) Newtons first law defines inertia and is rightly the force applied to it will be transmitted
called the law of inertia. Inertia are of three types : from string B to A through the mass M and
Inertia of rest, Inertia of motion and Inertia of as tension in A will be greater than in B by
Fig : 4.5
direction. Mg (weight of mass M), the string A will
(3) Inertia of rest : It is the inability of a body to break.
change by itself, its state of rest. This means a body at (v) If we place a coin on smooth piece of card
rest remains at rest and cannot start moving by its own. board covering a glass and strike the card board piece
Example : (i) A person who is standing freely in suddenly with a finger. The cardboard slips away and
bus, thrown backward, when bus starts suddenly. the coin falls into the glass due to inertia of rest.

When a bus suddenly starts, the force responsible (vi) The dust particles in a carpet falls off when it
for bringing bus in motion is also transmitted to lower is beaten with a stick. This is because the beating sets
part of body, so this part of the body comes in motion the carpet in motion whereas the dust particles tend to
along with the bus. While the upper half of body (say remain at rest and hence separate.
above the waist) receives no force to overcome inertia (4) Inertia of motion : It is the inability of a body
of rest and so it stays in its original position. Thus there to change by itself its state of uniform motion i.e., a
is a relative displacement between the two parts of the body in uniform motion can neither accelerate nor
body and it appears as if the upper part of the body has retard by its own.
been thrown backward. Example : (i) When a bus or train stops suddenly,
Note : (i) If the motion of the bus is slow, a passenger sitting inside tends to fall forward. This is
because the lower part of his body comes to rest with
the inertia of motion will be transmitted to the body of
the bus or train but the upper part tends to continue its
the person uniformly and so the entire body of the
motion due to inertia of motion.
person will come in motion with the bus and the person
will not experience any jerk. (ii) A person jumping out of a moving train may fall
(ii) When a horse starts suddenly, the rider tends forward.
to fall backward on account of inertia of rest of upper (iii) An athlete runs a certain distance before
part of the body as explained above. taking a long jump. This is because velocity acquired by
(iii) A bullet fired on a window pane makes a clean running is added to velocity of the athlete at the time of
hole through it, while a ball breaks the whole window. jump. Hence he can jump over a longer distance.
The bullet has a speed much greater than the ball. So (5) Inertia of direction : It is the inability of a
its time of contact with glass is small. So in case of body to change by itself it's direction of motion.
bullet the motion is transmitted only to a small portion
Example : (i) When a stone tied to one end of a
of the glass in that small time. Hence a clear hole is
string is whirled and the string breaks suddenly, the
created in the glass window, while in case of ball, the
stone flies off along the tangent to the circle. This is
time and the area of contact is large. During this time
because the pull in the string was forcing the stone to
the motion is transmitted to the entire window, thus
move in a circle. As soon as the string breaks, the pull
creating the cracks in the entire window.
vanishes. The stone in a bid to move along the straight
line flies off tangentially.
(ii) The rotating wheel of any vehicle throw out
mud, if any, tangentially, due to directional inertia.
(iii) When a car goes round a curve suddenly, the
person sitting inside is thrown outwards.
Cracks by the ball Hole by the bullet Newtons Second Law
Fig : 4.4
(iv) In the arrangement shown in the figure : (1) The rate of change of linear momentum of a
body is directly proportional to the external force
 Newton's Laws of motion 179
applied on the body and this change takes place
changes, speed remains
always in the direction of the applied force. constant. Force is always
perpendicular to velocity.
(2) If a body of mass m, moves with velocity v

then its linear momentum can be given by p mv and In non-uniform circular
v

motion, elliptical, parabolic
if force F is applied on a body, then or hyperbolic motion force
acts at an angle to the
Fd=p mg dp direction of motion. In all
F FK
dt dt these motions. Both
magnitude and direction of
dp
velocity changes.
or F (K = 1 in C.G.S. and S.I.
dt
units) (2) Dimension : Force = mass acceleration

2 2
d dv [F ] [M ][LT ] [MLT ]
or F (mv) m ma
dt dt (3) Units : Absolute units : (i) Newton (S.I.)
(ii) Dyne (C.G.S)
dv
(As a acceleration produced in the body) Gravitational units : (i) Kilogram-force (M.K.S.) (ii)
dt
Gram-force (C.G.S)

F ma
Newton : One Newton is that force which
Force = mass acceleration produces an acceleration of 1m/ s2 in a body of
Force mass 1 Kilogram.

(1) Force is an external effect in the form of a push 1 Newton 1kg m/ s2

or pull which Dyne : One dyne is that force which produces
(i) Produces or tries to produce motion in a body at an acceleration of 1cm/ s2 in a body of mass 1
rest. gram.
(ii) Stops or tries to stop a moving body. 1 Dyne 1gmcm/ sec2
(iii) Changes or tries to change the direction of
Relation between absolute units of force 1 Newton
motion of the body.
105 Dyne
Table 4.1 : Various condition of force application
Kilogram-force : It is that force which produces
F an acceleration of 9.8m / s2 in a body of mass 1
Body remains at rest. Here
force is trying to change kg.
u=0 v=0 the state of rest. 1 kg-f = 9.80 Newton
Gram-force : It is that force which produces an
F
Body starts moving. Here acceleration of 980cm/ s2 in a body of mass 1gm.
force changes the state of 1 gm-f = 980 Dyne
u=0 v>0 rest.
(4) F ma formula is valid only if force is
changing the state of rest or motion and the mass of the
In a small interval of time,
force increases the body is constant and finite.
F
u0 v>u magnitude of speed and (5) If m is not constant
direction of motion remains d
dv dm
same. F (mv) m v
dt dt dt
In a small interval of time,
(6) If force and acceleration have three component
F u force decreases the
along x, y and z axis, then
v<u magnitude of speed and
direction of motion remains F Fx i Fy and a
j Fzk ax
i ayj azk
same.
From above it is clear that
v In uniform circular motion Fx max , Fy may, Fz maz
F only direction of velocity
F
v
 180 Newton's Laws of Motion
(7) No force is required to move a body uniformly is zero or the work is path independent, the force is said
along a straight line with constant speed. to be conservative otherwise non conservative.

F ma F 0 (As a 0 ) Example : Conservative force : Gravitational force,
(8) When force is written without direction then electric force, elastic force.
positive force means repulsive while negative force Non conservative force : Frictional force, viscous
means attractive. force.
Example : Positive force Force between two (15) Common forces in mechanics :
similar charges (i) Weight : Weight of an object is the force with
Negative force Force between two opposite which earth attracts it. It is also called the force of gravity
charges or the gravitational force.
(9) Out of so many natural forces, for distance (ii) Reaction or Normal force : When a body is
10 15 metre, nuclear force is strongest while placed on a rigid surface, the body experiences a force
gravitational force weakest. which is perpendicular to the surfaces in contact. Then
Fnuclear Felectromag force is called Normal force or Reaction.
netic Fgravitatio
nal R
R
(10) Ratio of electric force and gravitational force
between two electrons Fe / Fg 10
43
Fe Fg

(11) Constant force : If the direction and mg cos
mg mg
magnitude of a force is constant. It is said to be a
Fig : 4.7 Fig : 4.8
constant force.
(iii) Tension : The force exerted by the end of taut
(12) Variable or dependent force :
string, rope or chain against pulling (applied) force is
(i) Time dependent force : In case of impulse or called the tension. The direction of tension is so as to
motion of a charged particle in an alternating electric pull the body. T=F
field force is time dependent.
(ii) Position dependent force : Gravitational force
Fig : 4.9
Gm1m2
between two bodies
r2 (iv) Spring force : Every spring resists any attempt
or Force between two charged particles to change its length. This resistive force increases with
change in length. Spring force is given by F Kx ;
q1q2
. where x is the change in length and K is the spring
4 0r 2 constant (unit N/m).

(iii) Velocity dependent force : Viscous force

(6rv)
Force on charged particle in a magnetic field
F = Kx
(qvBsin )
(13) Central force : If a position dependent force is
directed towards or away from a fixed point it is said to
be central otherwise non-central.
x
Example : Motion of Earth around the Sun. Motion of
Fig : 4.10
electron in an atom. Scattering of -particles from a
nucleus.
Equilibrium of Concurrent Force
Electro (1) If all the forces working on a body are acting on
F n the same point, then they are said to be concurrent.
+
Sun F - (2) A body, under the action of concurrent forces,
F + + particle is said to be in equilibrium, when there is no change in
Nucleus Nucleus
Eart the state of rest or of uniform motion along a straight
h line.
Fig : 4.6
(14) Conservative or non conservative force : If (3) The necessary condition for the equilibrium of
under the action of a force the work done in a round trip a body under the action of concurrent forces is that the
 Newton's Laws of motion 181
vector sum of all the forces acting on the body must be
zero.

(4) Mathematically for equilibrium
Fnet 0 or

Fx 0 ; Fy 0 ; , Fz 0
The table supports the book, by exerting an equal
(5) Three concurrent forces will be in equilibrium, force on the book. This is the force of reaction.
if they can be represented completely by three sides of
As the system is at rest, net force on it is zero.
a triangle taken in order. F
2 C Therefore force of action and reaction must be equal
B
and opposite.

F1 F3 (ii) Swimming is possible due to third law of

motion.
(iii) When a gun is fired, the bullet moves forward
A
Fig : 4.11
(action). The gun recoils backward (reaction)
(6) Lamis Theorem : For three concurrent forces in (iv) Rebounding of rubber ball takes place due to
third law of motion.
F1 F2 F
equilibrium 3
sin sin sin
F1
F2

R sin
R

F3
R cos
Fig : 4.12
Fig : 4.14
Newtons Third Law (v) While walking a person presses the ground in
the backward direction (action) by his feet. The ground
To every action, there is always an equal (in
pushes the person in forward direction with an equal
magnitude) and opposite (in direction) reaction.
force (reaction). The component of reaction in horizontal
(1) When a body exerts a force on any other body, direction makes the person move forward.
the second body also exerts an equal and opposite force (vi) It is difficult to walk on sand or ice.
on the first. (vii) Driving a nail into a wooden block without
(2) Forces in nature always occurs in pairs. A holding the block is difficult.
single isolated force is not possible. Frame of Reference
(3) Any agent, applying a force also experiences a (1) A frame in which an observer is situated and
force of equal magnitude but in opposite direction. The makes his observations is known as his Frame of
force applied by the agent is called Action and the reference.
counter force experienced by it is called Reaction.
(2) The reference frame is associated with a co-
(4) Action and reaction never act on the same ordinate system and a clock to measure the position
body. If it were so, the total force on a body would have and time of events happening in space. We can describe
always been zero i.e. the body will always remain in all the physical quantities like position, velocity,
equilibrium. acceleration etc. of an object in this coordinate system.
(5) If F AB = force exerted on body A by body B (3) Frame of reference are of two types : (i)
Inertial frame of reference (ii) Non-inertial frame of
(Action) and F BA = force exerted on body B by body A reference.
(Reaction) (i) Inertial frame of reference :
Then according to Newtons third law of motion (a) A frame of reference which is at rest or which is
moving with a uniform velocity along a straight line is
F AB F BA
called an inertial frame of reference.
(6) Example : (i) A book lying on a table exerts a
(b) In inertial frame of reference Newtons laws of
force on the table which is equal to the weight of the
motion holds good.
book. This is the force of action.
R

mg

Fig : 4.13
 182 Newton's Laws of Motion
(c) Inertial frame of reference are also called 1
unaccelerated frame of reference or Newtonian or F t
2
Galilean frame of reference.
(d) Ideally no inertial frame exist in universe. For
practical purpose a frame of reference may be (8) If Fav is the average magnitude of the force
considered as inertial if its acceleration is negligible then
with respect to the acceleration of the object to be t t
observed. I t 2 F dt Fav t 2dt Favt
1 1
(e) To measure the acceleration of a falling apple, (9) From Newtons second law F
earth can be considered as an inertial frame.
dp
(f) To observe the motion of planets, earth can not F
dt Fav
be considered as an inertial frame but for this purpose Impulse
the sun may be assumed to be an inertial frame. t2 p2
or t1 F dt p1 d pt t t2
t
Example : The lift at rest, lift moving (up or down) 1

with constant velocity, car moving with constant I p 2 p1 p
Fig : 4.16

velocity on a straight road.

i.e. The impulse of a force is equal to the change
(ii) Non-inertial frame of reference in momentum.
(a) Accelerated frame of references are called non- This statement is known as Impulse momentum
inertial frame of reference. theorem.
(b) Newtons laws of motion are not applicable in Examples : Hitting, kicking, catching, jumping,
non-inertial frame of reference. diving, collision etc.
Example : Car moving in uniform circular motion, In all these cases an impulse acts.
lift which is moving upward or downward with some I F dt Fav. t p constant
acceleration, plane which is taking off.
So if time of contact t is increased, average force
Impulse is decreased (or diluted) and vice-versa.
(1) When a large force works on a body for very (i) In hitting or kicking a ball we decrease the time
small time interval, it is called impulsive force. of contact so that large force acts on the ball producing
An impulsive force does not remain constant, but greater acceleration.
changes first from zero to maximum and then from
maximum to zero. In such case we measure the total (ii) In catching a ball a player by drawing his hands
effect of force. backwards increases the time of contact and so, lesser
(2) Impulse of a force is a measure of total effect force acts on his hands and his hands are saved from
of force. getting hurt.
t2
(3) I t1 F dt .
(4) Impulse is a vector quantity and its direction is
same as that of force.
(5) Dimension : [ MLT 1 ]

(6) Units : Newton-second or Kg-m- s1 (S.I.)

Fig : 4.17
Dyne-second or gm-cm- s1 (C.G.S.)
(iii) In jumping on sand (or water) the time of
(7) Force-time graph : Impulse is equal to the area contact is increased due to yielding of sand or water so
under F-t curve.
force is decreased and we are not injured. However if
If we plot a graph between force and time, the
we jump on cemented floor the motion stops in a very
area under the curve and time axis gives the value of
short interval of time resulting in a large force due to
impulse.
which we are seriously injured.
I Area between curve and time axis
(iv) An athlete is advised to come to stop slowly
Force

1 after finishing a fast race, so that time of stop increases

Base Height and hence force experienced by him decreases.
2 F
(v) China wares are wrapped in straw or paper
before packing.
t Tim
Fig : 4.15 e
 Newton's Laws of motion 183
Law of Conservation of Linear Momentum
If no external force acts on a system (called
isolated) of constant mass, the total momentum of the
system remains constant with time.
Let mG mass of gun, mB mass of bullet,
(1) According to this law for a system of particles

dp vG velocity of gun, vB velocity of

F
dt bullet

In the absence of external force F 0 then p Initial momentum of system = 0
constant
Final momentum of system mGvG mBvB
i.e., p p1 p2 p3 .... constant.

By the law of conservation of linear momentum
or m v m v m v .... constant
1 1 2 2 3 3 mGvG mBvB 0
This equation shows that in absence of external
force for a closed system the linear momentum of m
So recoil velocity vG B vB
individual particles may change but their sum remains mG
unchanged with time.
(a) Here negative sign indicates that the velocity
(2) Law of conservation of linear momentum is
of recoil vG is opposite to the velocity of the bullet.
independent of frame of reference, though linear
momentum depends on frame of reference. 1
(b) vG i.e. higher the mass of gun, lesser
(3) Conservation of linear momentum is equivalent mG
to Newtons third law of motion.
the velocity of recoil of gun.
For a system of two particles in absence of
(c) While firing the gun must be held tightly to the
external force, by law of conservation of linear
shoulder, this would save hurting the shoulder because
momentum.
in this condition the body of the shooter and the gun
p1 p2 constant. behave as one body. Total mass become large and recoil
velocity becomes too small.
m1v1 m2v2 constant.
1
Differentiating above with respect to time vG
mG mman
dv1 dv2
m1 m2 0 m1a1 m2a2 0 (iv) Rocket propulsion : The initial momentum of
dt dt
the rocket on its launching pad is zero. When it is fired
F1 F 2 0 from the launching pad, the exhaust gases rush
downward at a high speed and to conserve momentum,
F 2 F1 the rocket moves upwards. v
i.e. for every action there is an equal and opposite
m
reaction which is Newtons third law of motion.
(4) Practical applications of the law of
conservation of linear momentum
(i) When a man jumps out of a boat on the shore,
the boat is pushed slightly away from the shore.
(ii) A person left on a frictionless surface can get
away from it by blowing air out of his mouth or by
throwing some object in a direction opposite to the
u u
direction in which he wants to move.
(iii) Recoiling of a gun : For bullet and gun Fig : 4.19
system, the force exerted by trigger will be internal so
the momentum of the system remains unaffected.
Let m0 initial mass of rocket,

vG
vB

Fig : 4.18
 184 Newton's Laws of Motion
m = mass of rocket at any instant t The speed attained by the rocket when the
(instantaneous mass) complete fuel gets burnt is called burnt out speed of the
rocket. It is the maximum speed acquired by the rocket.
mr residual mass of empty container of the
rocket
Free Body Diagram
In this diagram the object of interest is isolated
u = velocity of exhaust gases,
from its surroundings and the interactions between the
v = velocity of rocket at any instant t object and the surroundings are represented in terms of
(instantaneous velocity) forces.
dm Example :
rate of change of mass of rocket = rate of T
T
dt a
a
fuel consumption m1 m2
= rate of ejection of the fuel.

dm
(a) Thrust on the rocket : F u mg
dt
Here negative sign indicates that direction of
thrust is opposite to the direction of escaping gases. R1 T R2
T
dm m1 a
F u (if effect of gravity is neglected) m1 m2 a
dt

u dm m1g cos m2g cos

(b) Acceleration of the rocket : a g m1g sin m2g sin
m dt Free body Free body
diagram of mass diagram of mass
m1 Fig : 4.20 m2
u dm
and if effect of gravity is neglected a Apparent Weight of a Body in a Lift
m dt
(c) Instantaneous velocity of the rocket : When a body of mass m is placed on a weighing
machine which is placed in a lift, then actual weight of
m the body is mg.
v uloge 0 gt R
m
and if effect of gravity is neglected

m m
v u loge 0 2.303u log10 0
m m
(d) Burnt out speed of the rocket : mg

m Fig : 4.21
vb vmax uloge 0
mr This acts on a weighing machine which offers a
reaction R given by the reading of weighing machine.
This reaction exerted by the surface of contact on the
body is the apparent weight of the body.

Table 4.2 : Apparent weight in a lift

Condition Figure Velocity Acceleration Reaction Conclusion

R mg = 0 Apparent weight
Lift is at rest v=0 a=0
LIFT R = mg = Actual weight

Spring
Balance

mg
 Newton's Laws of motion 185

Lift moving LIFT

upward or v = constant R mg = 0 Apparent weight
a=0
downward with R = mg = Actual weight
R
constant velocity

Spring
Balance

mg

Lift accelerating v = variable R mg = ma Apparent weight

LIFT
upward at the a<g
rate of 'a R = m(g + a) > Actual weight
R
a

Spring
Balance

mg
Lift accelerating v = variable R mg = mg Apparent weight
upward at the LIFT a=g
rate of g R = 2mg = 2 Actual weight
R
g

Spring
Balance

mg
Lift accelerating v = variable mg R = ma
LIFT Apparent weight <
downward at the a<g
R = m(g a) Actual weight
rate of a
R
a

Spring
Balance

mg
Lift accelerating Apparent weight
LIFT v = variable mg R = mg
downward at the a=g = Zero
rate of g R=0
(weightlessness)
R
g

Spring
Balance

mg Apparent weight
mg R = ma negative means the
Lift accelerating v = variable
LIFT body will rise from
downward at the a>g R = mg ma
the floor of the lift
rate of a(>g) R = ve
R and stick to the
a>g ceiling of the lift.

Spring
Balance

mg