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Vibrations and Waves > Oscillatory Motion > Flashcards

Oscillatory Motion Flashcards

1
Q

What causes periodic/oscillatory motion or vibration?

A

A net force towards the equilibrium position, a restoring force

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2
Q

Simple Harmonic Motion

A

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

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3
Q

Hooke’s Law

A

F = -kx

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4
Q

Simple Harmonic Motion

Displacement Equation

A

x = Acos(ωt+δ)

A = amplitude
δ = phase constant
ω = angular frequency 
t = time 
x = displacement
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5
Q

Time Period

Definition

A

Time taken for one complete oscillation

Time until the displacement is next the same again

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6
Q

Angular Frequency

Equation

A

ω = 2πf = 2π/T

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7
Q

Time Period for a Mass on a String

Equation

A

T = 2π √(m/k)

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8
Q

The Simple Pendulum

A

Simple Harmonic Motion with ω = √(g/L)

L = length of the string

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9
Q

What effects time period of a simple pendulum?

A

Time Period does not depend on mass or amplitude

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10
Q

Potential Energy in Simple Harmonic Motion

A

-PE is zero at the equilibrium position and maximum at amplitude

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11
Q

Potential Energy

Spring Compression Equation

A

U = kx²/2

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12
Q

Potential Energy

SHM Equation

A

Substitute x=Acos(ωt) into the PE of a spring equation

U = kA²cos²(ωt) / 2 = mω²x²/ 2

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13
Q

Kinetic Energy

SHM Equation

A

KE = mv²/2 = m(-ωAsin(ωt))²/2 = mω²A²sin²(ωt)/2

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14
Q

Total Energy in SHM

Description

A

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

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15
Q

Total Energy in SHM

Equation

A

E = KE + PE = mω²A²/2 = kA²/2

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16
Q

Modelling General Oscillations as SHM

A

At small enough amplitudes, any oscillation can be modelled as SHM

17
Q

Damped Oscillations

Description

A

All real oscillations are subject to dissipate forces which remove energy from the oscillating system and reduce the amplitude

18
Q

Damped Oscillations

Equation

A

d²x/dt² + γdx/dt + ω0²x = 0

γ = b/m
ω0² = k/m

Where b is the coefficient of resistance

19
Q

Solution to the Damped Oscillations Equation

A

x = A0 e^(-bt/2m) cos(ω’t + d)

20
Q

Damped Oscillations

b < 2mω0

A

Exponentially decaying oscillations, real solution

x = A0 e^(-bt/2m) cos(ω’t + d)

21
Q

Damped Oscillations

b = 2mω0

A

Critical damping, rapid return to the equilibrium position

x = (A+Bt)*e^(-ωt), no oscillations

22
Q

Damped Oscillations

b > 2mω0

A

Over damping, slow return to equilibrium

23
Q

Energy of a Damped Oscillator

Description

A

If amplitude decreases exponentially then energy (averaged per cycle) will also decrease exponentially
Since in SHM, E ∝ A²

24
Q

Energy of a Damped Oscillator

Equation

A

E = E0 * e^(-t/τ)

τ = decay time = m/b = 1/γ

25
Q

Quality Factor

Definition

A

Tells you how quickly the oscillations die away

A higher quality factor, the longer the time

26
Q

Quality Factor

Equation

A

Q = ω0τ

27
Q

Energy Loss Per Cycle

Equation

A

Δ|E| / E = 2πγ/ω0 = 2π/Q

28
Q

Driven Oscillations

Equation of Motion

A

d²x/dt² + γdx/dt + ω0²x = F0/m cos(ωt)

29
Q

Driven Oscillations

Solutions to the Equation of Motion

A

x = x0 cos(ωt - ϕ)

To determine x0 and ϕ, sub into equation of Motion for a driven oscillation

30
Q

Driven Oscillations and Resonance

A
  • 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
31
Q

Resonance and Damping

A

-when damping is applied to a resonating system, amplitude is decreased at every frequency and the frequency at which peak amplitude occurs is reduced

32
Q

Power Spectrum

A
  • 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

33
Q

Phase Lag

A

The amount by which displacement of the oscillator lags behind the driving force

ϕ = arctan{γω/(ω0²-ω²)}

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