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Chapter 18 Simple Harmonic Motion (18.6 (Explaining Resonance (When…
Chapter 18
Simple Harmonic Motion
18.1
Oscillations
Oscillations
- repeated motions about an equilibrium position
Equilibrium
- lowest point,point where object comes to rest
Amplitude
- maximum displacement
Free vibrations
- Constant amplitude, no frictional forces
Time Period
- time for 1 complete cycle of oscillation
Frequency
- number of cycles per second
Phase difference
Phase difference
- difference between 2 oscillating bodies of the same frequency (fraction if complete oscillation)
Phase difference (radians)= 2πΔt/T
18.3
Circles and waves
Circle
- y=rsinθ and x=rcosθ
Can use pendulum and wheel to show direct link between circular motion and SHM
If you set their periods to match, they'll have the same positions relative to each other
Always have same horizontal motion (velocity and acceleration)
Spring mass systems also show this but vertically
Sine wave solutions
Replace axis of displacement graph (cosθ) with amplitude (instead of max height) and frequency (instead of time period)
Gives equation
x=Acos(ωt)
if from max displacement
If released from equilibrium then replace with sin instead of cos
18.4
Mass-spring systems
Variables
2 things change time period
1)
Mass
2)
Spring constant
Positive displacement (up) then T<mg so resultant is down
Negative displacement (down) T>mg so resultant up
Weight constant so tension is restoring force
Simple pendulum
Resolving force comes from gravity- component that matters= one acting towards equilibrium=
mgsinθ
Resultant force= mgsinθ
Motion of bob only depends on length of pendulum
18.6
Forced vibrations
Periodic forces
- force applied at regular intervals
Natural frequency
- frequency system oscillates at with no periodic force
Forced vibrations
- vibrations you get when periodic force applied to system
Resonance
Reinforce
if periodic force works in direction of object- increase amplitude
Lower amplitude if periodic force works against oscillator
Finding Resonance
As periodic freq. increases amplitude increases until a maximum
After maximum amplitude decreases as you've passed resonant freq.
Explaining Resonance
When periodic force is applied, it's applied at wrong frequency- phase difference
Phase difference- force and motion won't synchronise
0-> π/2 amplitude increases π/2-> π amplitude decreases
Best phase difference= π/2, acceleration=0 and force in phase with velocity
Damping and resonance
Damping decrease frequency
Lighter the damping
- larger amplitude at resonance and closer resonant frequency is to natural frequency
Oscillating system with little/no damping at resonance . -
Frequency of periodic force= natural frequency of the system
18.5
Pendulum
Total energy= KE + GPE - at max. displacement all energy= GPE and at equilibrium all energy= KE
Mass-Spring
Total energy= Ep + Ek
Ek= 1/2 k (A^2 - x^2)
Damped oscillations
Dissipative forces
- forces reducing amplitude over time
Types
Critical
Just enough to stop system after it's released from a displaced position - oscillator returns to equilibrium in shortest time possible- in wind
Light
Oscillations slowly decrease in amplitude- T is constant- in air
Heavy
Returns to equilibrium more slowly than in critical, no oscillations occur- in oil
Energy-displacement graph
18.2
3 properties vary with time
Displacement
Velocity
Gradient of displacement time
Velocity greatest when gradient of displacement- time is greatest
Velocity zero when gradient of displacement-time graph is zero
Acceleration
Gradient of velocity-time graph
Acceleration greatest when gradient of velocity- time is greatest- velocity is zero and displacement is maximum in opposite direction
Acceleration zero when gradient of velocity-time graph is zero- here displacement is zero and velocity is maximum
Simple Harmonic Motion
2 things must be true
1)
acceleration proportional to displacement
2)
acceleration in opposite direction to displacement
a is proportional to -x
- gives equation
a=-ω^2 x
