PHY1014
Uniform Circular Motion
Any particle travelling at constant speed around a circle is
engaged in uniform circular motion.
The magnitude of 𝑣 is constant,
but since 𝑣 is everywhere
tangent to the circle, its direction
changes continuously.
v
r
v
r
O
r
In the particle model the centre
of the circle lies outside the particle,
v
and we speak of orbital motion.
Later we shall apply the same principles to the rotation or
spin of extended objects about axes within themselves.
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PHY1014
Period
The time taken for the particle to complete
one revolution (rev) is called the period, T,
of the motion.
Hence
v
v  2 r
T
r
O
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PHY1014
Angular Position
It will be more convenient to describe the position of an
orbiting particle in terms of polar coordinates rather than
xy-coordinates.
 , called the angular position of
the particle, …
y
r
s

• is positive when measured
O
x
counter-clockwise (CCW)
from the positive x-axis;
• is conveniently measured in radians (SI unit), where 1 rad
is the angle subtended at the centre by an arc length s = r;
• is the single time-dependent quantity of circular motion.
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PHY1014
Angular Position
Notes:
•
 (rad)  s
r
s = r
and
( in rad).
• The radian is a
dimensionless unit
(as is any unit of angle).
•
r
s

O
360  2 r rad  2 rad  1 rev
r
• 1 rad  360  57.3  60
2
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y
x
PHY1014
Angular Velocity
Change in angular position is
called angular displacement, .
Analogous to linear motion, the rate
of change of angular position is
called average angular velocity:
tf = t i + t
y
r
f
O

ti
r
i
x
average angular velocity   
t
Allowing t0, we get (instantaneous) angular velocity:
  lim    d
t  0  t
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dt
Units: [rad/s] (SI), but also
[°/s, rev/s, and rev/min  rpm]
PHY1014
Angular Velocity
Notes:
• A particle moves with uniform circular motion if and
only if its angular velocity  is constant.
•   2 rad , sign by inspection…
T
• Angular velocity is positive
for counterclockwise motion….
>0
• …negative for clockwise motion.
<0
• The graphical relationships we
developed for position s and velocity vs
in linear motion apply equally well to
angular position  and angular velocity …
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𝑑𝑠 𝑟𝑑𝜃
𝑑𝜃
𝑣𝑡 =
=
=𝑟
= 𝑟𝜔
𝑑𝑡
𝑑𝑡
𝑑𝑡
PHY1014
Velocity in Uniform Circular Motion
The velocity vector is tangent to the circle
v
r
v
r
O
r
ds rd
d
vt 

r
 r
dt
dt
dt
vr  0
vz  0
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v
Position  Velocity Graphs
PHY1014
Angular velocity  is equivalent to the slope of a -vs-t graph.
 (rad)
Eg:
2
0
–2
–4

2
4
6
8
t (s)
t
t

–2
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For the first 3 s the
slope
velocity
is     4  2   2 rad/s
 (rad/s)
0
A particle moves around a
circle…
2
4
6
8
t (s)
30
Position  Velocity Graphs
PHY1014
Angular velocity  is equivalent to the slope of a -vs-t graph.
 (rad)
Eg:
2
0
–2
2
4
6
8
t (s)
t
 (rad/s)

–2
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Between 3 s and 4 s the
slope
velocity
is   4   4   0 rad/s
–4
0
A particle moves around a
circle…
2
4
6
8
t (s)
1
Position  Velocity Graphs
PHY1014
Angular velocity  is equivalent to the slope of a -vs-t graph.
 (rad)
Eg:
2
0
–2
2
4
6
8
t (s)

–4
t
t

–2
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2
4
6
Between 4 s and 8 s the
slope
velocity
is   0   4    rad/s
 (rad/s)
0
A particle moves around a
circle…
8
t (s)
4
PHY1014
Finding Position From Velocity
A body’s angular position after a time interval t can be
determined from its angular velocity using
.
 f   i  t
Graphically, the change in angular position ( = t) is given by
the area “under” a -vs-t graph:
 (rad/s)
During the time
interval 2 s to 8 s the
body’s angular
displacement is
2


t
0
2
4
6
8
t (s)
t  2 rad/s   8  2  s
   12 rad
 i.e. 6 revs ccw 
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Example 1
PHY1014
The figure shows the angular-velocity-versus-time
graph for a particle moving in a circle. How many
revolutions does the object make during the first 4 s?
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Example 1
PHY1014
The angular position is the slope of the area under the ω vs. t graph
The area under the graph is 20 rad + 40 rad = 60 rad
Convert to revolutions 60 rad(1 rev/2 rad) = 9.5 rev
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PHY1014
The rtz-coordinate System
To facilitate the resolution of angular quantities, we introduce
the rtz-coordinate system (centred on the orbiting particle and
travelling around with it) in which…
• the r-axis (radial axis) points from the particle towards the
centre of the circle;
• the t-axis (tangential axis) is tangent to the circle, pointing in
the anticlockwise direction;
• the z-axis is perpendicular to the plane of motion.
z
z
r
t
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t
O
r
PHY1014
The rtz-coordinate System
Viewed from above
(with the z-axis
pointing out of the
screen) the axes are
shown travelling
around with the
particle…
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z
t
r
PHY1014
The rtz-coordinate System
Notes:
• As in xyz-coordinate system, the r-, t-, and z-axes are
mutually perpendicular.
• The rtz-coordinate system is used only to resolve vector
quantities associated with circular motion into radial
and tangential components.
The measurement of these quantities must necessarily
take place in other reference frames.
• Given some vector 𝐴 in the
plane of motion, making
an angle of  with the r-axis,
r
t
• Ar = A cos
A

• At = A sin
A cos
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A sin
PHY1014
Velocity and Angular Velocity
The velocity vector 𝑣 has
only a tangential component, vt .
t
vt
s = r
Differentiating with respect to time…

v t  ds  r d
dt
dt
Hence
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vt =  r
vt
and  
r
vr = 0
vt =  r
r
[m/s
rad
/s] 
 m 
vz = 0
vt
s
O
r
PHY1014
Velocity and Angular Velocity
Although the magnitude of 𝑣 remains constant in uniform circular
motion, its direction changes continuously, so the particle must be
accelerating.
Motion diagram analysis reveals
that the acceleration is centripetal.
a
Notes:
• For uniform circular motion, since
the lengths of successive Δ𝑣’sare all
the same, the magnitude of 𝑎 is constant.
• These are all average velocity vectors…
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a
a
a
vf
vi
v i
v
vf
PHY1014
Velocity and Angular Velocity
The instantaneous velocity and acceleration
vectors are everywhere at right angles to
each other.
During time interval t …
Q
v
Q'
• the particle travels an arc length vt
between P and P' (PP' ≈ vt);
• both the angular position and 𝑣 turn
through angles of  ;
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2

v
v


t
r
 P'
v
ar

O
P
ar
a
v
…so OPP' ||| P'QQ'
 v  v t
v
r
v
v
2

v
v
a  lim

r
t  0  t
PHY1014
Acceleration and Angular Velocity
In vector notation:

2
v
a
, towards centre of circle
r
And since v =  r…
t

vt
ar = 2r
r
Centripetal acceleration has
only a radial component, ar …
2
v
ar 
  2r
r
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at = 0
az = 0
O
ar
Example 2
PHY1014
Your roommate is working on his bicycle and has the bike
upside down. He spins the 60.0 cm diameter wheel, and you
notice that a pebble stuck in the tread goes by three times
every second.
a) What is the pebble's speed?
b) What is the pebble's acceleration?
The pebble’s angular velocity ω π = (3.0 rev/s)(2 rad/rev) 18.9
rad/s.
The speed of the pebble as it moves around a circle of radius
r = 30 cm
The radial acceleration is
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