• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Simple Harmonic Motion

Extracts from this document...

Introduction

Simple Harmonic Motion:

What is Simple Harmonic Motion

  • It is simple – not many forces
  • It is related to periodic waves
  • Its about motion (lol)

Example of SHM

  • A person in the ocean will experience SHM as waves go past.

image00.png

Example:

  • Hang a mass on a spring
  • Pull it down a little and let it go!

In common:

  • Oscillations: back and forth, up and down
  • Nice and stable: if you push it or pull it and let it go

You will get SHM if:

  1. The restoring force is directed towards the equilibrium position, i.e. goes back towards where it started
  2. The restoring force is proportional to the distance from the equilibrium. i.e. pull pendulum a long way, there is a bigger force.

e.g. pull the mass on the spring and the spring stretches and pulls back to where it normally hangs when you let it go.

WAVES

  • λ = wavelength (m)
  • T = period (s)
  • A or x0 = amplitude (m)
  • Period is the time it takes for ONE wave, frequency is the number of waves PER second

CIRCLES

  • Sine waves
  • Just expresses where they start periodically
  • f (frequency) is the symbol we use for how often history repeats
  • f is measured in hertz, Hz
  • f = 1/T or f-1
  • e.g. a frequency of 2Hz for a pendulum swinging means it swings twice in a second.

f = 1/T

ώ

  • ώ is “omega”
  • “angular frequency” or “angular velocity” (the speed it takes to go around a circle)
  • think of V = D / T
  • in wave motion:  ώ = d/t = 2(pi) / T = 2(pi)f
  • a sine wave can express a circle, this is just like saying you do one top of a circle (distance 2(pi)) in period (T)
  • ώ is really degrees per second.  

ώ = 2(pi) / T = 2(pi)f

image01.png

Radians

  • SI unit of an angle
  • 2(pi) radians = 360 degrees.

2(pi) radians = 360 degrees

More

  • to describe the motion of SHM we can use angles, say, for a pendulum
  • measuring the time is easier than angles
  • θ = ώt ; angle = degrees (or radians per second) x time

θ = ώt

The Math’s: Position

  • if you pull a weight on a string, or a pendulum, you can make SHM
  • Distance is ‘x’, amplitude is ‘x0
  • x(t) = Acosθ, but we want to know time so we use θ = ώt
  • x(t) = Acos(ώt) or just x = x0cos(ώt)
  • swings back and forth from A to –A
  • the period (T) is found by 2(pi)/ώ

x = x0cos(ώt)

Velocity

  • Just like in mechanics we sometimes want to know the velocity
  • Remember velocity = change in displacement / time
  • The gradient of a cosine graph is a negative sine graph
  • v(t) = -ώAsin(ώt) or just V = -v0sin(ώt)
  • where v0 = ώx0 – the maximum velocity (because it is maximum at the highest point)
...read more.

Middle

acceleration is the gradient of the velocity graphgradient of the sine graph is a cosine grapha(t) = -ώ2x0cos(ώt), but we know that x0cos(ώt) is x(t)Therefore, a(t) = -ώ2xMaximum value is at the extreme of displacementa0 = -ώ2x0

a = -ώ2x0

Mass on a Spring

  • if you hang a mass on a spring it will stretch the spring a bit and then just sit there
  • 2 forces: -         weight = mg (down)
  • spring = -kx (up)
  • “k” is the spring constant: if there is a big number, it is a tight spring, if there is a small number, it is a loose spring
  • it tells you how many Newton’s of force you will feel if you pull a spring ‘x’ meters from where it usually is.
  • If spring is not moving, the forces are equal at this point

Spring into Action

  • F = -kz, where k = mώ2

(Hooke’s Law)

v = -ώ√(x02 – x2)

SHM: Initial Conditions, (boundary conditions)

  • Remember that you can make a pendulum swing by
...read more.

Conclusion

image07.png
  • The waves get smaller
  • E.g. a pendulum with some friction or air resistance.
  • The amplitude slowly decreases with time.
  • However – the period does not change

Heavy (OVER) Damping

  • E.g. of a pendulum in water
  • Comes to a stop quickly

Critical Damping

  • E.g. pendulum in honey
  • NO oscillations

SHM: Natural Frequency (Resonant Frequency)

  • A person on a swing will swing back and forth at a particular frequency.  This is called the natural frequency of the swing.
  • Blowing across the top of the bottle makes a sound.  That is the natural frequency of the bottle
  • A guitar string plucked plays a particular note – its natural frequency
  • Every object has a natural frequency of vibration
  • E.g. a glass has a natural frequency
  • If you tap a glass you hear a sound.  The frequency of that sound is the glasses natural frequency
  • Normally the sound fades away due to damping.  If we introduce some FORCING we can keep it vibrating
  • E.g. pushing someone on a swing.
  • If you push at the same frequency the swing would naturally have, you get a big swing.  This is called RESONANCE.  
  • Forcing at an objects NATURAL FREQUENCY creates RESONANCEi.e. more bang for your buck.

In a Graph

image08.png

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Physics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Physics essays

  1. Investigate the factors affecting the period of a double string pendulum

    Distance l (cm) Frequency (Hz) ?0.21 2 0.61 4 0.76 6 0.89 8 1.02 10 1.11 12 1.23 In this graph, the trend line is polynomial and has a formula y=-0.001x2 + 0.082x +0.449 with a correlation of 0.998. However since the trend line based on the data points are

  2. Determine the spring constant of a vertical spiral spring in simple harmonic motion using ...

    at 60.00cm, Vertical Oscillation of 0.00cm Mass of 100g: Mass of 200g: Mass of 300g: Mass of 400g: Mass of 500g: Mass of 600g: - Data Table #3: The Spring Constant Mass Added �0.2g Oscillation of Spring �0.05 cm Fg (Fg=mg)

  1. Determining acceleration of free fall by of a simple pendulum.

    that the string of the pendulum may have extended when the pendulum was in motion, also the stand holding the pendulum could be dangling from side to side as the pendulum oscillates.

  2. Physics Wave revision question

    A piece of cork is floating in the water in the position shown. Which is the correct position of the cork a time later? (1) 27. A wave is travelling through a medium. The diagram shows the variation with time t of the displacement d of a particle of the medium from t = 0 to t = 25 ms.

  1. Simple Harmonic Motion Physics HL Lab Report

    accuracy, the string was cut at the marked point with a scissor. Data Collection Radius and length of the hook: Value of the smallest division on the scale (M): 0.1 cm Total no. of divisions on the vernier scale (n): 10 Least count (L.C)

  2. Finding the Spring Constant

    Since the mass keeps its force on the spring, it takes the spring a lot more time to complete an oscillation. The spring however cannot be indestructible because there is a certain limit of the spring where it cannot extend any further.

  1. Investigating the Oscillations of an Obstructed Pendulum

    There were definitely some sources of error, and these would definitely have had an effect on the results. These include both systematic and random errors. As can be seen from the graph on the previous page, not all the readings lie on the line of best-fit; some of the results lie on either side of the line.

  2. HL Physics Revision Notes

    Current experimental evidence suggests that low-frequency fields don?t harm genetic material. There has been evidence that this may lead to infant cancer and infant leukaemia. These risks are likely to be dependent on current (density) frequency, and length of exposure.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work