Simple Harmonic Motion

1
Simple Harmonic Motion
2
Harmonic Motion

• Linear Motion- A? B one place to another
• Harmonic Motion- Repeat over and over again ex.
  Swinging, walking in circles, pendulums, bicycle
  wheels etc.
• Cycle- building block of motion (back and forth
  one cycle)
• System all things we are interested in, exclude
  things we dont. Pendulum, system includes
  hanger, string and weight.
• Oscillator- any system that shows harmonic motion
  (heart, planets, pendulum etc.) Earth has
  several oscillating systems.

3
Harmonic Motion

• Light and sound come from oscillations
• Sound- oscillation of air(speaker pushes and
  pulls on air creating an oscillation of pressure)
• Color- light waves are electromagnetic
  oscillations. Faster oscillations make blue,
  slower make red.
• Technology- Fast electromagnetic oscillation
  (cell phones 100 million cycles/sec)
• (FM radio 88 to 107 million cylces/sec)

4
Harmonic Motion

• Period- time for 1 cycle
• Frequency- of cycles per second
• Hertz cycles/sec Hz
• t 1/f f 1/t inversely related
• Amplitude- size of cycle, can be distance or
  angle
• Damping- friction eventually slows down all
  oscillations and lowers amplitude

5
Frequency

• The FREQUENCY of a wave is the inverse of the
  PERIOD. That means that the frequency is the
  cycles per sec. The commonly used unit is
  HERTZ(HZ).

6
Pendulum Lab-only change 1 variable at a time!

• Part I
• Set up photogate so it can read as the ball
  passes through.
• Adjust things as many times as necessary but see
  what variable (amplitude, of washers or length
  of string has biggest effect on Period.
• Make sure timer is set on Period (double the time
  for 1 full cycle)
• Put results in data table and write conclusion.

• Part II
• Start at 30 degree amplitude, record time it
  takes to decrease down to 10 degree amplitude.
• Change mass 4 times, keeping string length
  constant.
• Change string length 4 times, keeping mass
  constant.
• Put results in data table and write conclusion.

7
Springs are like Waves and Circles
The amplitude, A, of a wave is the same as the
displacement ,x, of a spring. Both are in meters.
CREST
Equilibrium Line
Period, T, is the time for one revolution or in
the case of springs the time for ONE COMPLETE
oscillation (One crest and trough). Oscillations
could also be called vibrations and cycles. In
the wave above we have 1.75 cycles or waves or
vibrations or oscillations.
Trough
Tssec/cycle. Lets assume that the wave crosses
the equilibrium line in one second intervals. T
3.5 seconds/1.75 cycles. T 2 sec.
8
Simple Harmonic Motion

• Back and forth motion that is caused by a force
  that is directly proportional to the
  displacement. The displacement centers around an
  equilibrium position.

9
Springs Hookes Law

• One of the simplest type of simple harmonic
  motion is called Hooke's Law. This is primarily
  in reference to SPRINGS.

The negative sign only tells us that F is what
is called a RESTORING FORCE, in that it works in
the OPPOSITE direction of the displacement.
10
Hookes Law

• Common formulas which are set equal to Hooke's
  law are N.S.L. and weight

11
Example

• A load of 50 N attached to a spring hanging
  vertically stretches the spring 5.0 cm. The
  spring is now placed horizontally on a table and
  stretched 11.0 cm. What force is required to
  stretch the spring this amount?

110 N
1000 N/m
12
Hookes law practice problems

• 1. What force is necessary to stretch an ideal
  spring whose force constant is 120. N/m by an
  amount of 30. cm? 2. A spring with a force
  constant of 600. N/m is used on a scale for
  weighing fish. What is the mass of a fish that
  would stretch the spring by 7.5 cm from its
  normal length? 3. A spring in a pogo-stick is
  compressed 12 cm when a 40. kg girl stands on the
  stick. What is the force constant for the
  pogo-stick spring? 4. An elastic cord is
  80. cm long when it is supporting a mass of 10.
  kg hanging from it at rest at rest. When an
  additional 4.0 kg is added, the cord is 82.5 cm
  long.
• HINT 4 kg stretches the cord 2.5 cm!!
• (a) What is the spring constant of the
  cord?   (b) What is the length of cord when no
  mass is hanging from it?
• HINT once you have the k value, work the
  equation for spring force backwards!! THINK How
  much does 10.kg STRETCH the cord??5. A spring
  is connected to a wall. A mass on a horizontal
  surface is connected to the spring and pulled to
  the right along the surface stretching the spring
  by 25 cm. If the pulling force exerted on the
  mass was 80.N, determine the spring constant of
  the spring. You then hang an unknown mass hanging
  from the spring causes the spring to stretch 15
  cm, what is the mass of the unknown?

13
Hookes law lab

• Purpose Using the Springs and Swings apparatus,
  find the spring constant of each of 5 springs
  provided to you in lab. Label them with color
  and size.
• Procedure Placing a mass on the bottom hook of
  the spring, record how far down the spring moves
  mg k?x the mass of each mass is 12 g. You
  may use as many of the masses as needed.
• Record all data in a data table and label the k
  value for each spring.
• Find the unknown mass Using the information
  provided from your own data, find the mass of one
  individual magnet and of the plastic washer
  provided to your lab group. You may use any
  spring you wish, but you might want to use two or
  three to confirm your answer.
• Good Luck!

14
Hookes Law from a Graphical Point of View
Suppose we had the following data
x(m) Force(N)
0 0
0.1 12
0.2 24
0.3 36
0.4 48
0.5 60
0.6 72
k 120 N/m
15
We have seen F vs. x Before!!!!
Work or ENERGY FDx Since WORK or ENERGY is the
AREA, we must get some type of energy when we
compress or elongate the spring. This energy is
the AREA under the line!
Area ELASTIC POTENTIAL ENERGY
Since we STORE energy when the spring is
compressed and elongated it classifies itself as
a type of POTENTIAL ENERGY, Us. In this case,
it is called ELASTIC POTENTIAL ENERGY.
16
Elastic Potential Energy

• The graph of F vs.x for a spring that is IDEAL in
  nature will always produce a line with a positive
  linear slope. Thus the area under the line will
  always be represented as a triangle.

NOTE Keep in mind that this can be applied to
WORK or can be conserved with any other type of
energy.
17
Conservation of Energy in Springs
18
Example

• A slingshot consists of a light leather cup,
  containing a stone, that is pulled back against 2
  rubber bands. It takes a force of 30 N to stretch
  the bands 1.0 cm (a) What is the potential energy
  stored in the bands when a 50.0 g stone is placed
  in the cup and pulled back 0.20 m from the
  equilibrium position? (b) With what speed does it
  leave the slingshot?

3000 N/m
60 J
49 m/s
19
SHM and Uniform Circular Motion

• Springs and Waves behave very similar to objects
  that move in circles.
• The radius of the circle is symbolic of the
  displacement, x, of a spring or the amplitude, A,
  of a wave.

20
SHM and Uniform Circular Motion

• The radius of a circle is symbolic of the
  amplitude of a wave.
• Energy is conserved as the elastic potential
  energy in a spring can be converted into kinetic
  energy. Once again the displacement of a spring
  is symbolic of the amplitude of a wave
• Since BOTH algebraic expressions have the ratio
  of the Amplitude to the velocity we can set them
  equal to each other.
• This derives the PERIOD of a SPRING.

21
Example

• A 200 g mass is attached to a spring and executes
  simple harmonic motion with a period of 0.25 s If
  the total energy of the system is 2.0 J, find the
  (a) force constant of the spring (b) the
  amplitude of the motion

126.3 N/m
0.18 m
22
Pendulums

• Pendulums, like springs, oscillate back and forth
  exhibiting simple harmonic behavior.

A shadow projector would show a pendulum moving
in synchronization with a circle. Here, the
angular amplitude is equal to the radius of a
circle.
23
Pendulums
Consider the FBD for a pendulum. Here we have the
weight and tension. Even though the weight isnt
at an angle lets draw an axis along the tension.
q
mgcosq
q
mgsinq
24
Pendulums
What is x? It is the amplitude! In the picture
to the left, it represents the chord from where
it was released to the bottom of the swing
(equilibrium position).
25
Example

• A visitor to a lighthouse wishes to determine the
  height of the tower. She ties a spool of thread
  to a small rock to make a simple pendulum, which
  she hangs down the center of a spiral staircase
  of the tower. The period of oscillation is 9.40
  s. What is the height of the tower?

L Height 21.93 m