What does motion mean ?

Displacement of particle with respect to other particle is known as motion.

Types of Motion

Circular Motion By- Prof. Dnyanesh Vaidya

Circular motion Motion of a particle along the

circumference of a circle is called circular motion.

If the magnitude of this linear velocity (linear speed) remains constant, the body is said to be in uniform circular motion. (U.C.M.)

Position Vector or Radius Vector

A vector drawn from the centre of the circular path to the position of the particle performing circular motion at a instant, is called the position vector or radius vector of that particle performing circular motion at a instant

Angle BOA is called angular displacement.

Angular Displacement Angular displacement θ of a

particle in a given time is defined as the angle traced by its radius vector in that time at the origin of frame of reference.

∴ angular displacement = ∠BOA = θ

arcBABut

r

s

rIt is expressed in radian

,,,,,,,,,,,,,, s

r

Angular Velocity

The rate at which the angular displacement takes place is called angular velocity (ω)

,,,,,,,,,,,,,,,,,,,,,,,,,,,,

qvAverage angular velocityt

The instantaneous angular velocity can be obtained at limiting value of δt

,,,,,,,,,,,,,,,,,,,,,,,,,,,,

t 0

dlim

t dt

It is measured in rad/sec.

U.C.M.“ Motion of a body along the

circumference of a circle with uniform linear speed or

uniform angular velocity.”

Angular Acceleration

The rate of change of angular velocity is called ‘angular acceleration (α)’

2 1

av t

i 2t 0

d radlim It is measured in

t dt s

Multiple Choice Questions 1. In U.C.M. -------------- remain

constant (a) Velocity (b) linear velocity

(c) angular velocity (d) Magnitude of linear velocity.2. Determine the angular acceleration

of a rotating body which slows down from

500rpm to rest in 10 seconds.

(a) 5.233rad/s2 (b) - 5.233 rad/s2 (c) 5.233m/s2 (d) -5.233m/s2

Ans 1 : (d) a variable linear speed

2

2 12

1 1

2

2 : b 5.233 rad / s  

But 0

2

252 3.14

310

5.233 rad / s

tn

t t

Ans

Right Hand Rule

“ Imagine the axis of rotation to be held in your right hand with the fingers curled round the axis and the thumb out stretched along the axis. In this situation, if the curled fingers indicate the direction of rotation, the thumb indicates the direction of angular displacement, angular velocity and angular acceleration.

If the particle is moving in anticlockwise direction, the directions of angular displacement and velocity are along the axis in upward direction. The direction of angular acceleration is along the axis in upward direction if the angular velocity is increasing and downward if the angular velocity is decreasing.

Relation between Linear velocity and Angular velocity

According to the definition, velocity is distance travelled in unit time.

s

vtarc

angleradiuss

rs r

Instantaneous value of velocity can be obtained at the limiting value of δt as

t 0

rv lim

t

As r is not changing with respect to time, its limiting value is same

,,,,,,,,,,,,,,,,,,,,,,,,,, ,,

t

t

0

0

r  But, lim  is instantaneo

v r

us angular velocity .t

V vecto

v r limt

ral y, rl

v r

Relation between linear acceleration and angular acceleration

According to definition of linear acceleration, time rate of change of velocity is the magnitude of acceleration.

,,,,,,,,,,,,,,,,,,,,,,,,,, ,,

dvi.e. a

dt

da r

dtd

a rdt

But v r

But r is constant.

dBut angular acceleration

dt

a r Vectorally, a r

Acceleration of a body in U.C.M.

1. By Vector Method

Acceleration is time rate of change of velocity.

BR BQ QR ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

QR BR BQ ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

QR Velocity at B Velocity at A ,,,,,,,,,,,,,,

QR Change in velocity ,,,,,,,,,,,,,,

QR v ,,,,,,,,,,,,,,,,,,,,,,,,,,,,

According to the triangle law of vectors

,,,,,,,,,,,,,,v

accelerationdt

Instantaneous acceleration can be obtained by limiting δt to 0.

,,,,,,,,,,,,,,

t 0

va lim

t

∠QBR ≅ ∠AOB

∵ angle between the tangent is angle between the corresponding radii.

As δt approaches O, B approaches A and δθ becomes so small that we can write.

t 0

t 0

t 0

c

va lim

tBut V is constant 

 a v limt

But lim angular velocityt

a v

2

2c c

vSubstituting v r or ,

r

va r or a

r

Direction of this accelaration

3. In circular motion, the angle between the velocity and radial acceleration is :

(a) 0 (b) 90º (c) 180º (d) constantly changing

Ans : (b) 90º

2. By Calculus Method

The object moves from A to P in time t subtending angle θ at the centre Therefore

t

∴ θ = ωt

r xi y j

But t

r r i cos t j sin t

Time derivative of this position

vector will give us the linear velocity.

d di.e. v r r i cos t j sin t

dt dt

v r

r i r co

i sin t j c

s j r si

os

n

t

Time derivative of this velocity will give us acceleration.

n 2

n 2

n

n 2 2

n 2

0

dv di.e. accel r i sin t j cos t

dt dt

dvaccel r i cos t j sin t

dt

accel i r cos t j r sin t

But ( i r cos t j r sin t ) r

accel . r

accel r r

2 2c Ta a a

Home Work

1. The extremity of the hour hand of a clock moves 1/20 th as fast as the minute hand. What is the length of the hour hand if the minute hand if the minute hand is 10 cm long? ( vm = 20 vh , v = rw, Ans = 6 cm)

2. Find the speed at which the points on the equator move as the earth rotates about its axis. Radius of the earth = 6400 km. ( w = 2π/ T, v = rw, Ans = 465 m/s)