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Circular motion
Muhammad Imtiaz
Cs, Year 2
Namal College, Mianwali

Idea about circular motion
 Circular Motion :
 Motion of an object in a circular path is called circular motion.
 Circular motion is also known as:
Angular motion
Rotatory motion

Angular displacement
 The distance between two points in a circular path is measured by “ ”and is called
angular displacement.
 It is usually denoted by “ ”
 𝝦 = S / r
 It is a vector quantity.
 Units
“Radian(SI unit), Degree, Revolution”.
 Its direction is determined by RHR(right hand rule).
 Direction is +ve in counter clock wise direction
𝝦
𝝦

Relation of Radian, Degree
and revolution
 By completing one round:
 One round = one revolution
 One revolution = 2𝝅 rad
 2𝝅 rad = 360 degree
 1 rad = 360 / 2 𝝅 = 57.3 degree

Angular velocity
 Time rate of change of angular displacement is called angular velocity.
 It is denoted by “𝞈”
 𝞈 = Δ𝝦 / Δt
 Unit is “rad per sec (SI Unit)” , “revolution per minute”
 Direction is determined by RHR
 It is a vector quantity and direction will be along the axis of rotation
 Average angular velocity?
 Instantaneous angular velocity?

Angular acceleration
 Definition:
 Time rate of change of angular velocity is called angular acceleration.
 It is denoted by “α”
 α = Δ𝞈 / Δt
 Unit is [“rad/𝒔𝒆𝒄 𝟐
”] or [revolution/𝑚𝑖𝑛𝑢𝑡𝑒 𝟐
]
 It is a vector quantity direction will be along the axis of rotation
 Direction is determined by RHR
 Average angular acceleration?
 Instantaneous angular acceleration?

Relation b/w angular and linear velocity
 When an object moves in a circular path, its point “P” linear distance in time
interval “Δt” will be ΔS, while angular displacement will be Δ𝝦.
 ΔS = rΔ𝝦
 Dividing equation with Δt, we get
 V = r 𝞈

Centripetal force
 The force which keeps the body to move in a circular path is called centripetal
force.
 Fc = 𝐦𝒗 𝟐
/ r
 In angular motion
 Fc = mr𝒘 𝟐
 Centripetal acceleration:
 The acc. Of moving object in a circular path and whose direction is towards center is
called centripetal acceleration.
 Fc = mac

Moment of inertia
 The product of mass and square of radius is called moment of inertia.
 I = m𝑟2
 Unit is kg𝑚2
 It is scalar quantity
 Notes:
 It is rotational analogous of mass
 It depends upon mass and square of radius

Angular momentum
 The cross product of position vector ‘’r’’ and linear momentum “p” is called
angular momentum.
 L = r x p
 Its magnitude is “L = rpsin$”
 Its SI Unit is kg𝒎 𝟐
𝒔−𝟏
or Js
 It is a vector quantity
 Notes: if $ = 90
 L = rp = mvr As v = r𝞈
 L = m𝒓 𝟐
𝞈

Spin angular momentum
vs. orbital angular momentum
Spin angular momentum Orbital angular momentum
When an object about its own axis,
angular momentum is called spin
angular momentum
When an object moves about an
external axis/orbit, angular momentum
is called orbital angular momentum
It is represented by Ls It is represented by Lo
Radius is small, so body is considered a
point
Radius is larger than Ls

Law of conservation of angular momentum
 If no external torque is applied on the system, total angular momentum of the
system remain conserve.
 𝐼1 𝞈 1 = 𝐼2 𝞈 2

 It is a vector quantity.
 It direction is along the axis of rotation.
 e.g.
 Motion of earth around the sun.

Rotational K.E
 Definition:
 Energy gained by an object while moving in a circular path is called rotational K.E.
 K.Erot = ½ I𝞈 𝟐

Rotational K.E of disc and hoop
 For disc:
 I = ½ m 𝒓 𝟐
 So, K.E = ½ * (½ m𝒓 𝟐
) ∗𝞈 𝟐
 K.E = ¼ m𝒓 𝟐
𝞈 𝟐
As 𝒓 𝟐
𝞈 𝟐
= 𝑽 𝟐
 K.E = ¼ m𝒗 𝟐
 For hoop:
 I = m 𝒓 𝟐
 K.E = ½ * ( m𝒓 𝟐
) ∗𝞈 𝟐
 K.E = ½ m𝒓 𝟐
𝞈 𝟐
As 𝒓 𝟐
𝞈 𝟐
= 𝑽 𝟐
 K.E = ½ m𝒗 𝟐

Velocities of disc and hoop
 If both start moving down the inclined, their motion consist of both rotational and
translational motions. If there is no energy loss, then
 For disc
 For hoop

G.K.K / H.K.K
 Artificial satellites
 Real and apparent weight
 Weightlessness in satellites and gravity free system
 Orbital velocity
 Artificial gravity
 Geo stationary orbits
 Newton’s and Einstein views of gravitation

Fluid dynamics
 What is fluid?
 Anything which can flow is called fluid.
 E.g. all liquids and gases
 Fluid dynamics?
 The study of fluid in motion, called Fluid dynamics.
 Notes:
 This motion is describe by two laws.
 Law of conservation of mass gives us the equation of continuity.
 Law of conservation of energy is the basics of Bernoulli equation.

Viscosity, Drag force, Stokes law
 Viscosity:
 The frictional force between layers of fluid is called viscosity.
 It is denoted by Greek latter [η] “eata”.
 Drag force:
 Retarding force experience by object while moving through liquid.
 F = 6πηrv
 Which is called stokes law.

Terminal velocity
 Definition:
 When the magnitude of drag force becomes equal to the weight, the net force acting on
object zero. The constant speed at which it falls is called terminal velocity.
 Vt = 2g𝒓 𝟐
ρ / 9η

Fluid flow
Streamline / steady / laminar flow Turbulent flow
Flow of fluid in which every particle
flow the same path and passes through
the same point.
Irregular flow of fluid is called is
turbulent flow.

Equation of continuity
 Statement:
 The product of cross sectional area of pipe and fluid speed at any point is constant. This
constant is equals to volume flow per second or simply flow rate.
 Statement:
 A1V1 = A2V2

Bernoulli’s equation
 The sum of pressure, K.E per unit volume, P.E per unit volume is equals to constant.
 P + ½ ρ𝒗 𝟐
+ ρgh = constant


Applications of Bernoulli’s equation
 Torricelli theorem:
 Mathematical form:
 V2 = 𝟐𝒈(𝒉𝟏 − 𝒉𝟐)

Relation b/w speed and pressure of fluid
 Statement:
 Where the speed is high, pressure will be low.

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