Simple Harmonic Motion

1. Simple Harmonic Motion.

2. Revision.
For this part of the course you need to be
familiar with the concepts of circular motion and
angular velocity.
State the formulas used to calculate angular
velocity.

3. SHM
A system will oscillate if
there is a force acting on it
that tends to pull it back to
its equilibrium position – a
restoring force.
In a swinging pendulum the
combination of gravity and
the tension in the string that
always act to bring the
pendulum back to the centre
of its swing.
Tension
Gravity
Resultant
(restoring
force)

4. For a spring and mass the
combination of the gravity
acting on the mass and
the tension in the spring
means that the system will
always try to return to its
equilibrium position. M
T
g
Restoring
force

5. If the restoring force f is directly proportional to the
displacement x, the oscillation is known as simple
harmonic motion (SHM).
For an object oscillating with SHM
fµ -x
The minus sign shows that the restoring force is acting
opposite to the displacement.
Acceleration In SHM

6. DisplacementRestoring
force

7. Kinetic and Potential energy in SHM

8. We already know that f=ma, we can substitute this into
the above formula to give.
a µ -x
If we put in a constant we get the equation –
a = -w
2
x
w
2
is a constant, w is called the angular velocity and is
dependant on the frequency of the oscillations and can
be written as w=2pf.
So the equation for SHM can be written as:
a = -(2pf)
2
x

9. Questions.
•Use the equation a = -w2x to work out the correct units
of w.
•Sketch an oscillating pendulum and mark in the
positions of where the acceleration is greatest and
smallest.
•An ultrasonic welder uses a tip that vibrates at 25 kHz
if the tip’s amplitude is 6.25x10-2 calculate the
maximum acceleration of the tip

10. 1.A wave has a frequency of 4 Hz and an amplitude of
0.3m calculate its maximum acceleration.
3.A system is oscillating at 300 kHz with an amplitude of
0.6 mm calculate its maximum acceleration.
5.A speaker cone playing a constant bass note is
oscillating at 100 Hz, the total movement of the
speaker cone is 1 cm. Assuming that the movement is
SHM calculate its maximum acceleration.

11. Displacement.
If a system oscillates with SHM the pattern of the
motion will be the same, the motion will follow
the same rules.
The equation for the displacement, x, is
related to the time, t, by the equation:
x = Acoswt (remember to use radians)

12. We already know that w can also be written as
2pf, then the equation can also be written as:
x = Acos2pft
We can use this equation to predict the position
of an oscillating system at any time.

13. Qs
A pendulum is oscillating at 30 times per minute
and has an amplitude of 20 cm, find its
amplitude 0.5s after being released from its
maximum displacement.
Find its displacement 0.75s after being released
from its maximum displacement.
If the pendulum is oscillating at 120 times per
minute and has an amplitude of 30 cm.
Work out the displacement at 0.2 s, 0.4 s and
0.5 s

14. If high tide is at 12 noon and the next is 12 hours
later and the amplitude of the tide is 2m we can
work out the height of the tide at any time for
example 2pm.
Find out the height of the tide 2pm later (4pm).

15. Energy in simple harmonic motion.
We already know that the total energy
(mechanical energy) for a system that is moving
with SHM is the sum of the potential and kinetic
energies.
When the object is at the extremes of its
oscillation (x = ± A) it has no KE but the PE is at
its maximum. When the object is mid way
through its oscillation (when x = 0) the KE is at
its maximum but there is no PE.

16. The potential energy is equal to the work done.
Ep = ½ mw
2
x
2
At the maximum displacement E = Ep so
E = ½ mw
2
A
2
And
½ mw
2
A
2
= ½ mw
2
x
2
+ ½ mv
2
If we divide ½ m.
w
2
A
2
= w
2
x
2
+ v
2

17. So
v
2
= w
2
(A
2
- x
2
)
or
v = ±2pf ÖA
2
-x
2
We can use this equation to workout the velocity
of an object at any position in it’s SHM.

18. A 1 kg mass is hanging from a spring which has a spring
constant (k) of 1000 Nm-1, it is oscillating with an amplitude
of 2cm.
a)Use T = 2p Ö m/k to calculate the time period of the
oscillation.
b)use v= ±2pf ÖA
2
-x
2
to find the velocity and then the
kinetic energy of the mass.
-0.02
etc
0.01
0.015
0.02
KE, (joules)Velocity, (m/s)Displacement,
x (m)

19. Qs
A metal strip is clamped to the edge of the table
and has an object of mass 280g attached to the
free end. The object is pulled down and
released. The object vibrates with SHM with an
amplitude of 8.0 cm and a period of 0.16 s.
Calculate the maximum acceleration of the
object
Calculate the maximum force
State the position of the object when it has no
KE.

20. Describing SHM

21. http://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.html

22. Displacement
(x)
Acceleration (a)
p out of phase
(180 deg)
Velocity (v)
p/2 out of phase
(90 deg)

23. Simple pendulum
A pendulum consists of a
small “bob” of mass m,
suspended by a light
inextensible thread of
length l, from a fixed point.
The bob can be made to
oscillate about point O in a
vertical plane along the arc
of a circle.
We can ignore the mass of
the thread

24. We can show that oscillating simple pendulums
exhibit SHM.
We need to show that a µ x.
Consider the forces acting on the pendulum:
weight, W of the bob and the tension, T in the
thread.
We can resolve W into 2 components parallel
and perpendicular to the thread:
Parallel: the forces are in equilibrium
Perpendicular: only one force acts, providing
acceleration back towards O

26. Parallel to string:
F = mg cosq
Perpendicular: to string
F = restoring force towards O
= mg sinq
We already know that F = ma
So F= -mg sinq = ma
(-ve since towards O)

27. At small angles (q = less than 10 deg)
Sinq is approximately equal to q (in radians)
q approximates to x/l for small angles.
So: \ -mg (x/l) = ma
Rearranging: a = -g (x/l) = pendulum equation
(can also write this equation as a = -x (g/l))

28. In SHM a µ x
SHM equation a = -(2pf)2x
Pendulum equation a = -x (g/l)
Hence (2pf)
2
= (g/l)
\ f = 1/2p (Ög/l) (remember T = 1/f)
T = 2p (Öl/g)
The time period of a simple pendulum depends
on length of thread and acceleration due to
gravity

29. Experiment
Using a long clamp stand, a pendulum bob,
some light string and a stop watch to
investigate the relation ship between g, l
and T
For a pendulum of known length count the
time taken for 10 complete oscillations
(there and back).
Use the pendulum formula to calculate the
force of gravity.
Repeat with 3 other lengths.

30. SHM in Springs

31. In a spring-mass system.
Do you think the size of the mass affects the
Time Period of the Oscillation? What do you
think the relationship will be?
Do you think the stiffness (spring constant) of
the spring will affect the Time Period of the
Oscillation? What do you think the relationship
will be?

32. The diagram here shows a mass-
spring system.
Set the equipment up and use
F=Ke to determine the spring
constant of the spring.
Use 3 different masses to
determine the time period of
oscillation.
Double up the springs in parallel
and series and try to determine the
period of oscillation.
M

33. How did the different masses affect the period of
oscillation.
How did the spring constant (different
arrangements of springs) affect the period of
oscillation?

34. The time period of oscillation of a spring is
dependant on the spring constant of the spring
and the mass of the system.
It is independent of the force of gravity.
The relationship is.
÷
ø
ö
ç
è
æ
=
k
m
Tp2

35. Q1Calculate the time period of a spring mass
system of 2.5 kg with a spring const. of 200N/m.
Q2What is the frequency of a 20g mass oscillating
on the end of a spring with a const. of 120 N/m.
Q3A spring is oscillating 45 times per min.
calculate the mass if the const. is 1000 N/m.
Q4If a mass of 10000 Kg is oscillating at a
frequency of 0.37 Hz what is the const. of the
spring?

36. Resonance and damping.

37. Resonance.
Resonance is the tendency in a system to
vibrate at its maximum amplitude at a certain
frequency. This frequency is known as the
system's resonance frequency. When damping
is small, the resonance frequency is
approximately equal to the natural frequency of
the system, which is the frequency of free
vibrations.
The natural or fundamental frequency is often
written as f
0

38. Perhaps one of the more common examples of
resonance is in musical instruments. For
example in guitars it is possible to make other
strings vibrate “sympathetically” when another is
plucked, either at their fundamental or overtone
frequencies.
http://www.youtube.com/watch?v=MBZs5SCtlVA

39. Examples of resonance.
Pushing a child on a swing – maximum A when
pushing ¦ = ¦o
Tuning a radio – electrical resonance occurs
when ¦o of tuning circuit adjusted to match ¦ of
incoming signal
Pipe instruments - column of air forced to
vibrate. If reed ¦ = ¦o of column loud sound
produced
Rotating machinery – e.g. washing machine. An
out of balance drum will result in violent
vibrations at certain speeds

40. Tacoma narrows bridge.
The Tacoma narrows bridge is often used as an
example of resonance, although it is not strictly
scientifically accurate to do so. It does how ever
give an example of what can happen if an object
was to be kept at its resonant frequency for a
long time.
http://www.youtube.com/watch?v=j-zczJXSxnw&feature=related

41. Barton’s Pendulum
All objects have a natural frequency of vibration
or resonant frequency. If you force a system - in
this case a set of pendulums - to oscillate, you
get a maximum transfer of energy, i.e. maximum
amplitude imparted.
When the driving frequency equals the resonant
frequency of the driven system. The phase
relationship between the driver and driven
oscillator is also related by their relative
frequencies of oscillation.

43. You also get a very clear illustration of the phase
of oscillation relative to the driver. The pendulum
at resonance is π/2 behind the driver, all the
shorter pendulums are in phase with the driver
and all the longer ones are π out of phase.
The amplitude of the forced oscillations depend
on the forcing frequency of the driver and reach
a maximum when forcing frequency = natural
frequency of the driven cones.

44. Another example of resonance in a driven
system is the hacksaw blade oscillator.
Driving mass
And arm
‘Slave’ arm with
‘slave’ mass
Elastic
band
pointer

45. If we change the period of oscillation of the
driver by moving the mass (increasing L) the
hacksaw blade will vibrate at different rates, if
we get the driving frequency right the slave will
reach resonant frequency and vibrate wildly.
If we move the masses on the blade it will have
a similar effect.

46. Problems with resonance.
Resonance driver applies forces that continually
supply energy to oscillator ® increasing
amplitude.
A increases indefinitely unless energy
transferred away.
Severe case: A limit reached when oscillator
destroys itself. E.g. wine glass shatters when
opera singer reaches particular note.
How do we deal with unwanted resonance?
We could use damping, we could also change
¦o of object by changing its mass, if we were to
change the stiffness of supports (\moving
resonant ¦ away from driving ¦) we could reduce
the affect of resonance.

47. Damping
The amplitude depends on the degree of damping

48. A damped spring
Set up a suspended mass-
spring system with a
‘damper’ – a piece of card
attached horizontally to the
mass to increase the air
drag. Alternatively, clamp a
springy metal blade (e.g.
hacksaw blade) firmly to
the bench. Attach a mass
to the free end, and add a
damping card.
Show how the amplitude
decreases with time.