Transcript waves2 - World of Teaching

Waves
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Waves and Our Universe - Nelson
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Oscillations
1.
Going round in circles
2.
Circular Motion Calculations
3.
Circular Motion under gravity
4.
Periodic Motion
5.
SHM
6.
Oscillations and Circular Motion
7.
Experimental study of SHM
8.
Energy of an oscillator
9.
Mechanical Resonance
Waves
10. Travelling waves
11. Transverse and
Longitudinal waves
12. Wave speed, wavelength
and frequency
13. Bending Rays
14. Superposition
15. Two-source superposition
16. Superposition of light
17. Stationary waves
Going round in circles
• Speed may be constant
• But direction is continually
changing
• Therefore velocity is
continually changing
• Hence acceleration takes
place
Centripetal Acceleration
• Change in velocity is
towards the centre
• Therefore the
acceleration is
towards the centre
• This is called
centripetal
acceleration
Centripetal Force
• Acceleration is caused by
Force (F=ma)
• Force must be in the same
direction as acceleration
• Centripetal Force acts
towards the centre of the
circle
• CPforce is provided by
some external force – eg
friction
Examples of Centripetal Force
• Friction
• Tension in
string
• Gravitational
pull
Centripetal Force 2
What provides the cpforce in each case ?
Centripetal force 3
Circular Motion Calculations
• Centripetal
acceleration
• Centripetal
force
Period and Frequency
• The Period (T) of a body travelling in a circle
at constant speed is time taken to complete
one revolution - measured in seconds
• Frequency (f) is the number of revolutions per
second – measured in Hz
T=1/f
f=1/T
Angles in circular motion
• Radians are units of angle
• An angle in radians
= arc length / radius
• 1 radian is just over 57º
• There are 2π = 6.28
radians in a whole circle
Angular speed
T = 2π/ω = 1/f
f = ω/2π
• Angular speed ω is the
angle turned through
per second
• ω = θ/t = 2π / T
• 2π = whole circle angle
• T = time to complete
one revolution
Force and Acceleration
•
•
•
•
v = 2π r / T and
T = 2π / ω
v=rω
a = v² / r = centripetal acceleration
a = (r ω)² / r = r ω² is the alternative
equation for centripetal acceleration
• F = m r ω² is centripetal force
Circular Motion under gravity
• Loop the loop is
possible if the track
provides part of the
cpforce at the top
of the loop ( ST )
• The rest of the
cpforce is provided
by the weight of
the rider
Weightlessness
• True lack of weight can
only occur at huge
distances from any other
mass
• Apparent weightlessness
occurs during freefall
where all parts of you body
are accelerating at the
same rate
Weightlessness
These astronauts are in freefall
Red Arrows pilots
experience up to 9g (90m/s²)
This rollercoaster produces
accelerations up to 4g (40m/s²)
The conical pendulum
• The vertical component of the tension
(Tcosθ) supports the weight (mg)
• The horizontal component of tension
(Tsinθ) provides the centripetal force
Periodic Motion
• Regular vibrations or oscillations repeat the same
movement on either side of the equilibrium position f
times per second (f is the frequency)
• Displacement is the distance from the equilibrium
position
• Amplitude is the maximum displacement
• Period (T) is the time for one cycle or or 1 complete
oscillation
Producing time traces
• 2 ways of producing a voltage analogue
of the motion of an oscillating system
Time traces
Simple Harmonic Motion1
• Period is independent of
amplitude
• Same time for a large swing and
a small swing
• For a pendulum this only works for
angles of deflection up to about 20º
SHM2
• Gradient of
displacement v. time
graph gives a
velocity v. time graph
• Max veloc at x = 0
• Zero veloc at x = max
SHM3
• Acceleration v. time
graph is produced
from the gradient of
a velocity v. time
graph
• Max a at V = zero
• Zero a at v = max
SHM4
• Displacement and
acceleration are out
of phase
• a is proportional to - x
Hence the
minus
SHM5
• a = -ω²x equation defines SHM
• T = 2π/ω
• F = -kx eg a trolley tethered between two springs
Circular Motion and SHM
T = 2π/ω
• The peg following a circular path casts a
shadow which follows SHM
• This gives a mathematical connection
between the period T and the angular velocity
of the rotating peg