TOPIC 10: OSCILLATIONS Flashcards
[Definition] Oscillation
An oscillation is a to-and-fro motion between two limits
[Definition] Free Oscillations
Free oscillations are oscillations with constant amplitude AND without energy loss or gain as there is no driving or resistive forces acting on it
[Definition] Natural Frequency
Natural frequency is the frequency at which a body will vibrate when there is no driving or resistive forces acting on it
[Definition] Equilibrium Position
Quantities of Oscillation
Position where no net force acts on the oscillating mass
[Definition] Displacement, x
Quantities of Oscillation
Distance in a specified direction from equilibrium position of oscillating mass
[Definition] Amplitude, x₀
Quantities of Oscillation
Maximum distance from equilibrium position
[Definition] Period, T
Quantities of Oscillation
Time taken for one complete oscillation of the oscillating mass
[Definition] Frequency, f
Quantities of Oscillation
Number of complete oscillations per unit time
[Definition] Phase, ϕ
Quantities of Oscillation
An angular measure of the fraction of a cycle completed by the oscillating mass
[Definition] Phase Difference, Δϕ
Quantities of Oscillation
Measure of how much an oscillation is out of step with another oscillation
[Definition] Angular frequency
Quantities of Oscillation
Defined as the product of 2π and frequency
ω = 2πf (units: radian s⁻¹)
[Definition] Simple Harmonic Motion
SHM is a type of oscillatory motion where the acceleration is directly proportional to the displacement from the equilibrium position and directed opposite to displacement
[Formula] Main Formulas of Oscillation
Variation with x
a = - ω²x
v = ±√(x₀² - x²)
[Formula] Main Formulas of Oscillation
Variation with t
If x = 0
x = x₀ sin(ωt)
v = ωx₀ cos(ωt)
a = - ω²x₀ sin(ωt)
If x = x₀
x = x₀ cos(ωt)
v = - ωx₀ sin(ωt)
a = - ω²x₀ cos(ωt)
Finding equations for horizontal spring-mass systems
Combine with topic on Forces
F = ma = kx
a = - ω²x
to find ω, f, T etc
Finding equations for vertical spring-mass systems
Combine with topic on Forces
F = ma
kx = ma
a = - ω²x
to find ω, f, T etc
Finding equations for simple pendulum
Combine with topic on Forces
F = ma
mg sinθ = ma
a = - ω²x
to find ω, f, T etc
Finding energy of oscillations
- Always find kinetic energy first
- Followed by total energy (when Ek max)
- Potential energy
Same for both (variation with displacement / time)
[Definition] Damped Oscillations
Damped oscillations are oscillations in which the amplitude decreases with time as a result of dissipative forces that reduces the total energy of the oscillations
Light to Critical to Heavy Damping
Light to Critical to Heavy Damping:
Light: More time needed to return to equilbrium
Critical: Shortest possible time (no oscillation)
Heavy: More time than critical (no oscillation)
[Definition] Forced Oscillations
Forced oscillations are oscillations where there is a continuous input of energy by an external periodic force that maintains the oscillation amplitude
Amplitude - driving frequency graph
Increases from non-zero y-intercept to max (at natural frequency), then decreases to near zero
[Definition] Resonance
Resonance is a phenomenon in which the amplitude of an oscillatory motion is at maximum because there is maximum rate of transfer of energy from the external driver to the oscillating system.
This occurs when the driving frequency of the external periodic force equals to the natural frequency.
Amplitude - driving frequency graph of no damping to increased damping
- Max amplitude decreases
- Max amplitude shifts to left (cos period longer, so frequency decreases)
- Flatter curve
