Oscillatory Motion Flashcards
What causes periodic/oscillatory motion or vibration?
A net force towards the equilibrium position, a restoring force
Simple Harmonic Motion
A special case of oscillatory Motion where the restoring force is proportional to the displacement but acts in the opposite direction to displacement
This means that the acceleration of the object is also proportional to its displacement but in the opposite direction
Hooke’s Law
F = -kx
Simple Harmonic Motion
Displacement Equation
x = Acos(ωt+δ)
A = amplitude δ = phase constant ω = angular frequency t = time x = displacement
Time Period
Definition
Time taken for one complete oscillation
Time until the displacement is next the same again
Angular Frequency
Equation
ω = 2πf = 2π/T
Time Period for a Mass on a String
Equation
T = 2π √(m/k)
The Simple Pendulum
Simple Harmonic Motion with ω = √(g/L)
L = length of the string
What effects time period of a simple pendulum?
Time Period does not depend on mass or amplitude
Potential Energy in Simple Harmonic Motion
-PE is zero at the equilibrium position and maximum at amplitude
Potential Energy
Spring Compression Equation
U = kx²/2
Potential Energy
SHM Equation
Substitute x=Acos(ωt) into the PE of a spring equation
U = kA²cos²(ωt) / 2 = mω²x²/ 2
Kinetic Energy
SHM Equation
KE = mv²/2 = m(-ωAsin(ωt))²/2 = mω²A²sin²(ωt)/2
Total Energy in SHM
Description
The total energy remains constant assuming no resistive force
At equilibrium kinetic enemy is maximum and potential energy is 0
At amplitude kinetic energy is 0 and potential energy is maximum
The total of KE and PE at any point is a constant
Total Energy in SHM
Equation
E = KE + PE = mω²A²/2 = kA²/2
Modelling General Oscillations as SHM
At small enough amplitudes, any oscillation can be modelled as SHM
Damped Oscillations
Description
All real oscillations are subject to dissipate forces which remove energy from the oscillating system and reduce the amplitude
Damped Oscillations
Equation
d²x/dt² + γdx/dt + ω0²x = 0
γ = b/m ω0² = k/m
Where b is the coefficient of resistance
Solution to the Damped Oscillations Equation
x = A0 e^(-bt/2m) cos(ω’t + d)
Damped Oscillations
b < 2mω0
Exponentially decaying oscillations, real solution
x = A0 e^(-bt/2m) cos(ω’t + d)
Damped Oscillations
b = 2mω0
Critical damping, rapid return to the equilibrium position
x = (A+Bt)*e^(-ωt), no oscillations
Damped Oscillations
b > 2mω0
Over damping, slow return to equilibrium
Energy of a Damped Oscillator
Description
If amplitude decreases exponentially then energy (averaged per cycle) will also decrease exponentially
Since in SHM, E ∝ A²
Energy of a Damped Oscillator
Equation
E = E0 * e^(-t/τ)
τ = decay time = m/b = 1/γ
Quality Factor
Definition
Tells you how quickly the oscillations die away
A higher quality factor, the longer the time
Quality Factor
Equation
Q = ω0τ
Energy Loss Per Cycle
Equation
Δ|E| / E = 2πγ/ω0 = 2π/Q
Driven Oscillations
Equation of Motion
d²x/dt² + γdx/dt + ω0²x = F0/m cos(ωt)
Driven Oscillations
Solutions to the Equation of Motion
x = x0 cos(ωt - ϕ)
To determine x0 and ϕ, sub into equation of Motion for a driven oscillation
Driven Oscillations and Resonance
- amplitude and energy of the system in the streaky state depends on amplitude and frequency of the driver
- without a driving force, a system will oscillate at its natural frequency
- when a driving force is applied at the same frequency as the natural frequency the system resonates and amplitude tends to infinity
Resonance and Damping
-when damping is applied to a resonating system, amplitude is decreased at every frequency and the frequency at which peak amplitude occurs is reduced
Power Spectrum
- when damping is weak, the oscillator absorbs much more energy from the driving force than it loses, the resonance peak is tall and narrow
- when damping is stronger, the resonance curve is lower and broader
For weak damping, Δω/ω0 = Δf/f0 ~ 1/Q
Phase Lag
The amount by which displacement of the oscillator lags behind the driving force
ϕ = arctan{γω/(ω0²-ω²)}
