Chapter 5

Newton’s Laws
of Motion

FROM FIRST LAW OF MOTION

A body continues to be in its state of rest or in uniform motion along a straight
line unless an external force is applied on it.
This law is also called law of intertia.
Example: If a moving vehicle suddenly stops, then the passengers inside the
vehicle bend outward.

FROM SECOND LAW OF MOTION

dPx dPy dPz
Fx
= = ma x ; = Fz =
Fy = ma y ; = ma z
dt dt dt

FROM THIRD LAW OF MOTION

 
FAB = −FBA

FAB = Force on A due to B

FBA = Force on B due to A
Force
Force is a push or pull which changes or tries to change the state of rest, the
state of uniform motion, size or shape of body.
Its SI unit is newton (N) and its dimensional formula is [MLT–2].
Forces can be categorised into two types:
(i) Contact Forces: Frictional force, tensional force, spring force, normal
force etc are the contact forces.
(ii) Distant Forces: (Field Forces) Electrostatic force, gravitational force,
magnetic force etc. are action at a distance forces.
31
Weight (w)
It is a field force. It is the force with which a body is pulled towards the centre
of the earth due to gravity. It has the magnitude mg, where m is the mass of
the body and g is the acceleration due to gravity.
w = mg
Weighing Machine
A weighing machine does not measure the weight but measures the force
exerted by object on its upper surface.
Normal Reaction
It is a contact force. It is the force between two surfaces in contact, which is
always perpendicular to the surfaces in contact.
Tension
Tension force always pulls a body.
Tension is a reactive force. It is not an active force.
Tension across a massless pulley or frictionless pulley remains constant.
Rope becomes slack when tension force becomes zero.
Spring Force
F = –kx
x is displacement of the free end from its natural length or deformation of the
spring where K = spring constant.
Spring Property
K ×  = constant
where  = Natural length of spring
If spring is cut into two in the ratio m : n then spring constant is given by
m n
= 1 = ; 2
m+n m+n
k = k11 = k22
For series combination of springs
1 1 1
= + +
k eq k1 k 2

For parallel combination of spring
keq = k1 + k2 + k3 …
Spring Balance
It does not measure the weight. It measures the force exerted by the object
at the hook.
Hand Book (Physics) 32
Remember
V1 + V2
VP =
2
a1 + a 2
aP =
2

VP

V1
V2

(m 2 − m1 )g
a= T T
m1 + m 2 a m1
a
2m1m 2 g
T=
m1 + m 2    m2

WEDGE CONSTRAINT
Components of velocity along perpendicular direction to the contact plane of
the two objects is always equal if there is no deformations at contact place
and they remain in contact.

VP

V1
V2 V3
V1 sin 
 Contact 
V3 = V1 sin  Plane
33 Newton’s Laws of Motion
NEWTON’S LAW FOR A SYSTEM
   
Fext = m1a1 + m 2 a 2 + m3 a 3 + 
Fext = Net external force on the system.
m1, m2, m3 are the masses of the objects of the system and a1, a2, a3 are the
acceleration of the objects respectively.
Equilibrium of a Particle
When the vector sum of the forces acting on a body is zero, then the body is
said to be in equilibrium.
O
F2 F1
If two forces F1 and F2 act on a particles, then they will be in equilibrium if
F1 + F2 = 0.
Strategy for solving problems in static equilibrium
• Determine all the forces that are acting on the rigid body. They will
come from the other objects with which the body is in contact (supports,
walls, floors, weights resting on them) as well as gravity,
• Draw a diagram and put in all the information you have about these
forces: The points on the body at which they act, their magnitudes
(if known), their directions (if known).
• Write down the equations for static equilibrium. For the torque equation
you will have a choice of where to put the axis: in making your choice
think of which point would make the resulting equations the simplest.
• Solve the equations! (That’s not physics... that’s math.) If the problem
is well-posed you will not have too many or too few equations to find
all the unknowns.
FRICTION
Friction force is of two types–(a) Kinetic, (b) Static.
Kinetic Friction
fk = mk × N
The proportionality constant mk is called the coefficient of kinetic friction and
its value depends on the nature of the two surfaces in contact.
Static Friction
It exists between the two surfaces when there is tendency of relative motion
but no relative motion along the two contact surfaces.
This means static friction is a variable and self adjusting force. However it
has a maximum value called limiting friction.
fmax = µSN
0 < fS < fsmax
Hand Book (Physics) 34
 n
tio
Friction

ric
cF
ati
µS N
µKN

St
f M F (effort)
(friction) Applied Force
µ S , µR
Angle of Friction
It is the angle which the resultant ofo the force of limiting friction and the
normal reaction (N) makes with the direction of N.
R N


f = N

ml = tan q

N

Angle of Repose or Angle of Sliding

=
N

fk
It is the minmum angle of inclination of a plane
with the horizontal, such that a body placed on it,
just begins to slide down.
mg cos


If angle of repose is a and coefficient of

sin

mg
limiting friction is ml , then
g


m

ml = tan a
Pseudo Force
When an observer is on an accelerating frame of reference, the observer will
measure acceleration on another mass without any external force.
If a0 is acceleration of observer and he measures the pseudo force FP on
rest mass m, the magnitude of pseudo FP = ma0 and its direction is opposite
to direction of ovserver.


FP = ma0 a0

35 Newton’s Laws of Motion

CIRCULAR MOTION
Definition of Circular Motion
When a particle moves in a plane such that its distance from a fixed
(or moving) point remains constant then its motion is called as circular
motion with respect to that fixed point. That fixed point is called centre and
the distance is called radius of circular path.
Radius Vector
The vector joining the centre of the circle and the centre of the particle
performing circular motion is called radius vector. It has constant magnitude
and variable direction. It is directed outwards.
Frequency (n or f)
Number of revolutions described by particle per sec. Its unit is revolutions
per second (r.p.s.) or revolutions per minute (r.p.m.).
Time Period (T)
It is time taken by particle to complete one revolution.

1
T=
n
arc length s
• Angle q = = (Unit → radian)
radius r
∆θ
• Average angular velocity w= (a scalar) unit → rad/sec
∆t
dθ
• Instantaneous angular velocity w = (a vector) unit → rad/sec
dt

r
 s
r

2π
• For uniform angular velocity w= = 2pf or 2pn
T
• Angular displacement q = wt

Hand Book (Physics) 36

 v
• Relation between w (uniform) and v ω=
r
  
• In vector from velocity v = ω× r
  
 dv d   dω   dr
• Acceleration =a = (ω× =
r) × r + ω×
dt dt dt dt
     
= α × r + ω× v = a t + a C
dv
• Tangential acceleration: at = = ar
dt
v2 
• Centripetal acceleration : aC = wv = = w2r or ˆ
a C = ω2 r (r)
r
2 2
 v 2   dv 
• Magnitude of net acceleration : a = a C2 + a 2t =   + 
 r   dt 
Maximum/Minimum Speed in Circular Motion

• On unbanked road : vmax = µs Rg

 µ + tan θ 
• On banked road : v max =  s  Rg
 1 − µs tan θ 

(tan θ − µS )Rg
=v min v min ≤ v car ≤ v max
1 + µS tan θ

where f = angle of friction = tan–1ms; q = angle of banking.

v2
• Bending of cyclist : tan q =
rg

N

2
ac = v /r

mg
37 Newton’s Laws of Motion
 KEY TIPS

• Average angular velocity is a scalar physical quantity whereas

instantaneous angular velocity is a vector physical quantity.

• Small Angular displacement dθ is a vector quantity, but large angular
displacement q is scalar quantity.
       
dθ1 + dθ2 = dθ2 + dθ1 But θ1 + θ2 ≠ θ2 + θ1
Relative Angular Velocity
Relative angular velocity of a particle ‘A’ w.r.t. other moving particle B is
the angular velocity of the position vector of A w.r.t. B.

vA cos 1

1
A vA

vB
r
vA sin 1

2
vB sin 2
B
That means it is the rate at which position vector of ‘A’ w.r.t. B rotates at
that instant
(v AB ) ⊥ Relative velocity of A w.r.t. B perpendicular to line AB
ω
=AB =
rAB separation between A and B
here (vAB)^ = vA sin q1 + vB sin q2
v sin θ1 + v B sin θ2
\ ωAB =A
r

qqq

Hand Book (Physics) 38