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Simple harmonic motion

– a type of periodic motion, where the restoring force is directly proportional to the displacement,
and acts in the direction opposite to the displacement .

The pendulum of a grandfather clock exhibits simple harmonic motion.

old-grandfather-clock

A mass on a spring has SHM.

Water in a tube

What is a restoring force?

(TBA)

SHM Restoring force balls on springs\

(tba)

Convert circular motion into linear SHM

The red ball has circular motion.

The blue point is the shadow of the red ball (shown on the X-axis.)

The shadow’s motion is SHM.

shm-circular-motion-to-linear-motion-gif

Here’s an object moving up and down.

Graph it’s height (Y-position) as a function of time.

Pendulums

A pendulum is a weight suspended from a pivot so that it can swing freely.

They were first used scientifically in 1602 by Galileo Galilei. Became a standard way to make clocks (timekeeping).

The pendulum clock invented by Christian Huygens in 1658 became the world’s standard timekeeper, used in homes and offices for 270 years.

Some achieved accuracy of about one second per year before it was superseded by quartz clocks in the 1930s.

The period of a pendulum gets longer as the amplitude θ0 (width of swing) increases.

A pendulum, showing the velocity and acceleration vectors.

Animations and apps

Compadre SHM interactive apps

Flash Animations for Physics, David M. Harrison

Flash Interactive, PCCL

See these example oscillation apps

27. Simple gravity pendulum

28. Simple gravity pendulum | MCQ

29. Simple gravity pendulum|Velocity & force vectors

PowerPoint: Giancoli Vibrations and Waves SHM Chap 11

Topics to be added later

vibration / oscillation

periodic

equilibrium position

Hooke’s Law F = – kx

displacement

amplitude

cycle

period’

frequency

leads to simple harmonic motion

this system is also called a simple harmonic oscillator

Energy in the simple harmonic oscillator

11th grade physics

To stretch or compress a spring work has to be done. Hence potential energy is stored in the stretched or compressed spring.

PE = 1/2kx^2

The total mechanical energy of the mass-spring system is the sum of the kinetic and potential energy

E = 1/2mv^2 + 1/2kx^2

the total mechanical eenrgy of this system is proportional to the square of the amplitude

The period and sinusoidal nature of SHM

11th grade physics

The period of a SHM does not depend on the amplitude.

[more to be added]

The simple pendulum

A simple pendulum consists of a small object (the “bob”) suspended from the end of a lightweight cord. We assume that the cord doesn’t stretch, and that it’s mass is small compared to that of the bob.

[formulas inserted here]

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11th grade physics

A pendulum is a weight suspended from a pivot so that it can swing freely.

When displaced sideways from its equilibrium position, it is subject to a restoring force due to gravity

Gravity accelerates it back toward the equilibrium position.

When released, the restoring force combined with the pendulum’s mass causes it to oscillate about the equilibrium position.

The time for one complete cycle, a left and right swing = period.

The period depend on: length of the pendulum (for small swings)

Wow – so the swing’s period is (approximately) the same for different size swings?!  Amplitude of the swing (or equivalently, the angle, theta) doesn’t matter?! Cool. Thsi is called isochronism – it is why pendulums are so useful for timekeeping.

For large swing, though, a fuller analysis shows that is also depends on the amplitude. In that case the physics gets more complicated, and we’ll skip over that here.

First well analyzed in 1602 by Galileo Galilei

Used for timekeeping. Was the world’s most accurate timekeeping technology until the 1930s.

Used in accelerometers and seismometers.

Used to be used as gravimeters to measure the acceleration of gravity in geophysical surveys.

Ethymology: from the Latin pendulus, meaning ‘hanging’.

The simple gravity pendulum is an idealized mathematical model of a pendulum.

We have a weight (“bob”) on the end of a massless cord, suspended from a pivot, without friction, or air resistance.

Real pendulums are subject to friction and air drag, so the amplitude of their swings declines over time,

Text above adapted from https://en.wikipedia.org/wiki/Pendulum

Animation of a pendulum showing forces acting on the bob: the tension T in the rod and the gravitational force mg.

Animation of a pendulum showing the velocity and acceleration vectors.

Math

The period depends on the pendulum’s length, and g, the acceleration of gravity

For small swings, where  θ0 is small, we can derive a simple formula (below)

Notice that the mass of the bob doesn’t appear!

Here is how to calculate the period, T (one complete cycle)

5aa04824df4e09c1ae352502bdee9c92

Damped Harmonic Motion

[text to be added here]

Forced vibrations and resonance

[more text to be added here]

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Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media.

Introductory Physics, High School Learning Standards for a Full First-Year Course, Mass 2006
4. Waves carry energy from place to place without the transfer of matter.
4.1 Describe the measurable properties of waves (velocity, frequency, wavelength, amplitude, period) and explain the relationships among them. Recognize examples of simple harmonic motion

SAT subject test in Physics: Waves and optics
 General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect

AP Physics learning standards
F. Oscillations and Gravitation
1. Students should understand simple harmonic motion, so they can: Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period and frequency of the motion.

Giancoli Physics outline: Oscillations and Waves

  • 11.1: Simple Harmonic Motion—Spring Oscillations
  • 11.2: Energy in Simple Harmonic Motion
  • 11.3: The Period and Sinusoidal Nature of SHM
  • 11.4: The Simple Pendulum
  • 11.5: Damped Harmonic Motion
  • 11.6: Forced Oscillations; Resonance
  • 11.7: Wave Motion
  • 11.8 Types of waves – transverse and longitudinal
  • 11.9: Energy transported by waves
  • 11.10: Intensity related to amplitude and frequency
  • 11.11 Reflection and transmission
  • 11.12 Interference: the principle of superposition
  • 11.13: Standing waves and resonance
  • 11.14: Refraction
  • 11.15: Diffraction
  • 11.16: Mathematical representation of a travelling wave
  • resource: http://www.acoustics.salford.ac.uk/feschools/waves/shm.php#motion
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