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Circular Motion
PHYA 4
Further Mechanics
How can we make an object travel
in a circle?
• Hint: think about Newton’s 1st law...
Circular motion
• Remember Newton’s 1st law?
– an object will remain at rest or in uniform motion in a
straight line unless acted upon by an external force
• So what is needed to make something go
around in a circle?
– A resultant force
• Remember Newton’s 2nd law?
– F=ma
• So a body travelling in a circle constantly
experiences a resultant force (and is
accelerated) towards the centre of the circle
– This is not an equilibrium situation! An unbalanced
force exists!
A bucket of water on a rope
• If we spin the bucket fast enough in a
vertical circle, the water stays in the
bucket
– Why?
A mass on a string
• Speed of rotation remains constant
• Velocity is constantly changing, so mass is
constantly accelerating towards centre of circle
• So there is a constant force on the mass
towards the centre of the circle
– Tension in string (until you let go!)
Talking about circular motion
The radian
Rotation and speed
• No gears, so as the pedals
are turned, the wheel goes
round with them with a
period T
• The wheel rim is travelling
faster than the pedals,
although both are rotating
at the same frequency, f
2r
• Speed of rim:
v
T
 2rf
So the speed an
object moves depends
on the frequency of
rotation and the
radius
Talking about circular motion
• Angular displacement (q) = no. of radians
turned through
• Angular speed (w) = no. of radians turned
through per second
2
s
w  2f 
speed
T
dq s / r v
r
w

 , or v  wr
dt
t
r
(sometimes called angular velocity)
Worked example: Calculating w
A stone on a string: the stone moves round
at a constant speed of 3 ms-1 on a string of
length 0.75 m
• What is the instantaneous linear speed of
the stone at any point on the circle?
• What is the angular speed of stone at any
point on the circle?
Practice Questions
• Examples 1: Radians and angular speed
Centripetal acceleration
• Acceleration directed towards centre
– Centripetal means “centre seeking”
• Size depends on:
– How sharply the object is turning (r)
– How quickly the object is moving (v)
dv
a
dt
vector
Centripetal acceleration
s v
q   , so v  qv
r
v
v qv
a

t t
but w 
q
,
t
so a  wv.
Remember v  wr ,
object
2
v
so a  w 2 r 
r
Centripetal Force
F  ma,
2
mv
2
F
 mw r
r
• Force acts towards the centre of the circle,
not outwards!
• Not a special type of force
Examples of sources of centripetal
force
Planetary orbits
gravitation
Electron orbits
electrostatic force on electron
contact force (reaction) at the
walls
Centrifuge
Gramophone needle
the walls of the groove in the
record
Car cornering
friction between road and tyres
Car cornering on
banked track
component of normal reaction
Aircraft banking
horizontal component of lift on the
wings
Worked Example: Centripetal Force
A stone of mass 0.5 kg is swung round in a
horizontal circle (on a frictionless surface) of
radius 0.75 m with a steady speed of 4 ms-1.
Calculate:
(a) the centripetal acceleration of the stone
(b) the centripetal force acting on the stone.
No such thing as centrifugal force...
• Centrifugal means “centre fleeing”
• It is an “effective force” you feel when in a
rotating frame of reference
• e.g., cornering car
No such thing as centrifugal force...
• Car applies a
force towards the
centre of the
circle
• Driver feels a
force pushing
him outwards
– Reaction force
• Physics joke...
Practice Questions
• Centripetal force sheet
• Whirling bung experiment
• Examples sheet 2
Hump-backed bridges
• Centripetal force provided
by gravity
• Above a certain speed, v0,
this force is not enough to
keep vehicle in contact
with road
2
0
mv
mg 
so v 0  gr
r
Note: independent of mass...
Roundabouts and corners
• What provides the
centriptal force?
– Friction
• What factors affect
the maximum speed a
vehicle can corner?
– Radius of corner
– Limiting frictional force
2
0
mv
F0 
r
( mg)
: coefficient of friction (not examinable)
Banked tracks
• On a flat road, only friction
provides the centripetal
force
– Above a certain speed you
lose grip
• On a banked track there is a
horizontal component of the
reaction force towards the
centre of the curve
– No need to steer! (at least at
one particular speed)
Optimum speed on a banked track
• Can you derive an expression for the
speed at which no steering is required for
a circular track of radius r, banked at an
angle q?
Banked tracks – speed for no
sideways friction
• Resolving reaction
force horizontally
and vertically:
mv2
n sin q 
r
n cos q  mg
2
v
2
• so tan q 
, or v  gr tan q
gr
Wall of death
Ball of death
Speed at which a
vehicle can travel
around a banked curve
without steering
Fairgrounds
• Many rides derive their excitement
from centripetal force
– A popular context for exam questions!
– Read pages 26-29
– Answer questions on p.29
Simple Harmonic Motion
PHYA 4
Further Mechanics
Oscillations in nature
• Oscillation is nature’s way of finding equilibrium
• This interplay can be found throughout nature:
–
–
–
–
–
–
–
–
–
A swinging pendulum
Waves on water
A plucked string (and the eardrum of a listener)
Vibrating atoms in a lattice
Voltages and currents in electric circuits
Excited electrons emitting light
A bouncing ball
Ocean tides
Populations of predators and prey in an ecosystem...
Simple Harmonic Motion
• Harmonic motion: motion that repeats
itself after a cycle
• Simple: simple!
• Let’s look at some examples...
displacement
time
velocity
time
acceleration
time
• Displacement/velocity/acceleration
animation
• x/v/a Java applet
Simple Harmonic Motion Summary
• What is SHM?
• What sort of systems display SHM?
• How can we describe SHM?
• What is happening to the energy of an
ideal system undergoing SHM?
Displacement of mass on a spring
Mass on spring terminology
When do you get SHM?
• A system is said to oscillate with SHM if
the restoring force:
– is proportional to the displacement from
equilibrium position
– is always directed towards the equilbrium
position
Force, acceleration, velocity and displacement
Phase differences
Time traces
varies with time like:
displacement s
/2 = 90
/2 = 90
 = 180
cos 2ft
... the velocity is the rate of change
of displacement...
–sin 2ft
... the acceleration is the rate of
change of velocity...
–cos 2ft
...and the acceleration tracks the force
exactly...
–cos 2ft
velocity v
acceleration = F/m
same thing
zero
If this is how the displacement varies
with time...
force F = –ks
displacement s
... the force is exactly opposite to
the displacement...
cos 2ft
Mass on spring Energy transfer
Mass on spring Energy
SHM is like a 1D projection of
uniform circular motion
Phasors
• A rotating vector which represents a wave
• Length corresponds to amplitude, angle
corresponds to phase
Damping
• In a real system there is always some
energy loss to the surroundings
• This leads to a gradual decrease in the
amplitude of the oscillation
– For light damping, the period is
(approximately) unaffected, though.
• The damping force generally is linearly
proportional to velocity
– Resulting in exponential decrease of
amplitude
Damping
Damping example
Under-damping
Critical Damping
• Critical damping provides the quickest approach to zero amplitude
Over-damping
Damping summary
• An underdamped oscillator approaches
zero quickly, but overshoots and oscillates
around it
• A critically damped oscillator has the
quickest approach to zero.
• An overdamped oscillator approaches
zero more slowly.
What’s going on here?
• Example 2
Free and Forced vibration
• When a system is displaced from its equilibrium
position it oscillates freely at its natural
frequency
– No external force acts
– No energy is transferred
• When an external force is repeatedly applied the
system undergoes forced oscillation
– energy is transferred to the system.
• Eg Barton’s pendulums
Resonant driving
Resonance
• If the system happens to be driven at its
natural frequency the transfer of energy is
most efficient: this is RESONANCE
– Oscillation is positively reinforced every cycle
– Amplitude quickly builds up
• Resonance can lead to uncontrolled,
destructive vibrations
– Bridges, glasses and opera singers, etc.
Amplitude vs driving frequency
Effect of damping on resonance
Further investigation
• Pendulum lab
• Masses on springs