Download circular motion

CIRCULAR MOTION
Angular Motion
•
•
•
•
Angular displacement: 
d
Angular velocity:   dt
Angular acceleration   d
dt
Uniformly accelerated motion
Linear Vs Angular Kinematics
Linear Motion
Angular Motion
Relationship
(r = radius)
Quantity
Unit
Quantity
Unit
s
m

rad
s = r
v
m s-1

rad s-1
v = r
a
m s-2

rad s-2
a = r
• Period: T
• Frequency: f
2

 2f
T
Relation between Tangential and
Angular Velocities
v  r
Uniform Circular Motion
• Tangential acceleration:
v(cos   1)
at  lim
0
t 0
t
v2
2
 r
• Centripetal (Normal) acceleration:
r
Centripetal Force
• A resultant force acting towards the centre
• Centripetal acceleration
• Centripetal force:
2
mv
2
F
 mr
r
Conclusion
• Not a new type of force
• Force  velocity
• Centripetal force does not imply the object
will move to the centre of the circle
• Experimental verification
• The force does no work on the object
• If the force ceases to act, the object will
move off tangentially
Experimental Verification
Computer simulation
Examples of Circular Motion
• Orbital motion of satellites and heavenly
bodies
• Spinning of machine parts or wheels
• Motion of charged particles in a magnetic
field
• Early models of atoms
Further Examples
• Turning of a vehicle round a corner
• Bicycle turning in a smooth banked track
• Liquid spinning in a bucket about a vertical
axis
• Aircraft turning in flight
Conical Pendulum
l cos 
Period T  2
g
Motion of Cyclist Round Circular
Track
• Condition for skidding:
tan  > 
•  is independent of m
• In turning a sharp
corner,  must be large
Motion of Car round Circular
Track
1
v2h
R1  m( g 
)
2
ra
1
v 2h
R2  m( g 
)
2
ra
•Car will overturn if v 
•Car will skid if
gar
h
v  gr
Banking
• For no side-slip at the
wheels
v2
tan  
gr
• Daily example: racing
track
Aircraft Turning in Flight
• Banking angle for the turn:
v2
tan  
gr
Centrifuge
• To separate
particles in
suspension from the
less dense liquid
• Procedure
Rotor
• The person will not
slip down if

g
r
Variation of g with Latitude
• g’ = g - r2
Motion in a Vertical Circle
• Ring threaded on a smooth vertical circular
wire [Figure]
• Suspended particle in a vertical circle
[Figure]
• The outside of a smooth vertical circular rod
[Figure]
Conditions of Describing a
Complete Vertical Circle
• Case I: the particle is suspended by a light
rigid rod v0  2 gl
• Case II: the particle is suspended by a light
string v0  5gl
[Figure]
Bucket of Water Whirled in a
Vertical Circle
• For the water to stay in the bucket: v 
gr
Looping the loop
• To describe a complete circle:
h  5r/2
Examples
Orbits
Back
Turning Round a Corner
Centripetal force is provided by the frictional force
between the wheels and the road
Back
Banked Track in Cycling
Centripetal force is
provided by the
horizontal component of
the normal reaction.
Back
Ring Threaded on a Smooth
Vertical Circular Wire
Back
Suspended Particle in a Vertical
Circle
Back
The Outside of a Smooth Vertical
Circular Rod
Back
Conditions for Describing a
Complete Vertical Circle
Back