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# Simple Harmonic Motion

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Introduction

Simple Harmonic Motion:

What is Simple Harmonic Motion

• It is simple – not many forces
• It is related to periodic waves

Example of SHM

• A person in the ocean will experience SHM as waves go past. 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 • SI unit of an angle
• 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)

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

Conclusion • 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.

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