Oscillations Flashcards
What is free oscillation
A body is said to undergo free oscillation when the only external force acting on it is the restoring force. It oscillates at its natural frequency with constant amplitude and no loss of energy
What is frequency
f = 1/T
What is angular frequency
ω = 2πf = 2π/T
What is phase
θ = ωt = (2π/T)t
What is phase difference
Phase difference Φ is an angular measure of the fraction of a cycle that 2 oscillations having the same frequency are out of step
What is simple harmonic motion
SHM is an oscillatory motion of a particle whose acceleration is proportional to its displacement from the equilibrium position and is always directed towards that position
a = -ω^2x
To prove that a motion is simple harmonic, it is sufficient to establish that aα-x
If the particle starts at equilibrium position at t=0,
x = x₀sinωt
v = ωx₀cosωt
- ωx₀ = v₀ = vmax
a = -ω²x₀sinωt
- ω²X₀ = a₀ = amax
If the particle starts at amplitude at t=0,
x = x₀cosωt
v = ωx₀sinωt
- ωx₀ = v₀ = vmax
a = -ω²x₀cosωt
- ω²X₀ = a₀ = amax
Velocity in terms of discplacement in SHM
v = ±ω√(x₀²−x²)
Energy in SHM variation with time
K.E. = ½mv² = ½mω²sin²ωt P.E. = ½kx² = ½mω²x² = ½mω²x₀²cos²ωt E.T. = ½mω²x₀²
Energy in SHM variation with displacement
K.E. = ½mω²(x₀²-x²) P.E. = ½mω²x² T.E. = ½mω²x₀²
Energy in SHM
Maximum P.E. = Maximum K.E. = ½mω²x₀² = T.E.
What are damped oscillations
A damped oscillation is an oscillation that loses energy due to dissipative forces
The total energy and amplitude decreases exponentially
What is light damping
It is when the dissipative force is small so that definite oscillations still occur but the amplitude of oscillation decreases exponentially with time
What is critical damping
It is when the dissipative force is of a critical value so that the system returns to equilibrium position in the shortest time possible without oscillation
What is heavy damping
It is when the dissipative force is large so that the system returns very slowly to equilibrium position without oscillation
What is natural frequency
Characteristic frequency at which a system oscillates when it is set into motion
What is driving frequency
It is the frequency of the external periodic force which replenishes the energy of an oscillating system
What is forced oscillation
It occurs when an oscillating system is subjected to an external periodic force. It oscillates at the driving frequency of the applied force
What is resonance
Resonance occurs when the driving frequency of the external periodic force equals the natural frequency of the oscillating system. There is maximum energy transfer and the system oscillated with maximum amplitude.
What are the effects of damping on resonance
- Amplitude at all frequency is reduced and the peak becomes flatter or broader
- Maximum amplitude occurs at a lower driving frequency than the natural frequency
