Introduction to the subject Motion

2.  What does motion mean ?

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

4. Types of Motion

5. Circular Motion
By- Prof. Dnyanesh Vaidya

6. Circular motion
 Motion of a particle along the
circumference of a circle is called
circular motion.

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

12. 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

13. Angle BOA is called angular displacement.

14. 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
arcBA
But
r
s
r
It is expressed
in radian
 s
r

15. Angular Velocity
 The rate at which the angular
displacement takes place is called
angular velocity (ω)


qvAverage angular velocity
t
The instantaneous angular velocity can be obtained at limiting
value of δt


t 0
d
lim
t dt
It is measured in rad/sec.

16. U.C.M.
 “ Motion of a body along the
circumference of a circle with
uniform linear speed or uniform
angular velocity.”

17. Angular Acceleration
 The rate of change of angular velocity
is called „angular acceleration (α)‟
2 1
av
t
i 2t 0
d rad
lim It is measured in
t dt s

18.  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

19.  Ans 1 : (d) a variable linear speed

20. 2
2 1
2
1 1
2
2 : b 5.233 rad / s
But 0
2
25
2 3.14
3
10
5.233 rad / s
t
n
t t
Ans

21. 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.

22.  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.

23. Relation between Linear
velocity and Angular velocity

24.  According to the definition, velocity is
distance travelled in unit time.
s
v
t
arc
angle
radius
s
r
s r
Instantaneous value of velocity can be obtained at
the limiting value of δt as
t 0
r
v lim
t

25.  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 lim
t
ral y, rl

26. v r

27. Relation between linear acceleration
and angular acceleration
According to definition of linear
acceleration, time rate of change of velocity
is the magnitude of acceleration.
  
dv
i.e. a
dt
d
a r
dt
d
a r
dt
But v r
But r is constant.
d
But angular acceleration
dt
a r Vectorally, a r

28. Acceleration of a body in U.C.M.
 1. By Vector Method

29.  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
acceleration
dt

30.  Instantaneous acceleration can be
obtained by limiting δt to 0.

t 0
v
a lim
t
∠QBR ≅ ∠AOB

31.  ∵ 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.

32. t 0
t 0
t 0
c
v
a lim
t
But V is constant
a v lim
t
But lim angular velocity
t
a v
2
2
c c
v
Substituting v r or ,
r
v
a r or a
r

33. Direction of this accelaration

34.  3. In circular motion, the angle
between the velocity and radial
acceleration is :
(a) 0 (b) 90º
(c) 180º (d) constantly changing

35.  Ans : (b) 90º

36. 2. By Calculus Method
The object moves from A to P in time t subtending angle θ at the
centre Therefore
t
∴ θ = ωt

37. 
  
  
   
 
 

r xi yj
But t
r r i cos t j sin t
Time derivative of this position
vector will give us the linear velocity.
d d
i.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

38.  Time derivative of this velocity will give
us acceleration.
 

 
 
 




n 2
n 2
n
n 2 2
n 2
0
dv d
i.e. accel r i sin t j cos t
dt dt
dv
accel 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

39. 2 2
c Ta a a

40. 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)