Simple Harmonic Motion

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

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)

V = -v0sin(ώt)

Acceleration

• acceleration is the gradient of the velocity graph
• gradient of the sine graph is a cosine graph
• a(t) = -ώ2x0cos(ώt), but we know that x0cos(ώt) is x(t)
• Therefore, a(t) = -ώ2x
• Maximum value is at the extreme of displacement
• a0 = -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 pulling it to one side and letting it go.
• It STARTS at x0 : x = - x0cos(ώt)

• You can also make a pendulum swing by pushing it when it is hanging down
• It STARTS at 0 ; x = x0sin(ώt)

• These are (pi)/2 or 90° out of phase. They are the same graph, it is the same motion, but the choice of when you start timing, is different

Two Sets of Equations

• ώ = 2(pi)f or ώ2 = k / m
• v0 = ώx0

SHM: Energy

• remember if you drop an object there is a conservation of energy: it is the same everywhere.  The total energy is conserved.

• With SHM we cant rely on just using MGH for Ep. mgh is only for gravitational potential energy
• Work = change in energy = Force x Displacement
• W = ΔE = Fx
• In SHM F = -kx
• If you graph F vs. x
• The area under the line is the energy
• Area = 0.5bh (of triangle in graph)

 = 0.5 (x) (F)

 = 0.5 (x) (kx)

 = 0.5kx2

• Ep = 0.5kx2

• Ep = 0.5mώ2x2

Kinetic Energy

• Just 0.5mv2
• Ek = 0.5m[ώ √(x02 – x2)]2
• Ek = 0.5mώ2(x02 – x2)

• Et = Ep + Ek

• Et = 0.5mώ2x02

• If you graph Ep and Ek:

• NOTE: -         Ep: Maximum at x0 (furthest away)

                    Minimum at 0 (equilibrium position)

• Ek: Maximum at 0 as it zooms past

       Minimum at x0 as it stops before swinging back

• Et: conserved, always the same

SHM: Dampening (Stopping it from being perfect)

• So far friction has been ignored, due to SIMPLE harmonic motion
• Although no maths is involved here, terminology will be looked at.

Light (UNDER) Damping

• 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 RESONANCE i.e. more bang for your buck.

In a Graph