what to take note for displacement?
displacement is always from the equilibrium position
describe free oscillations.
an oscillating system when displaced from its equilibrium position oscillates about this position with a natural frequency of the system.
when undamped this free oscillation system has constant amplitude and no energy is lost to surroundings.
what are the 2 formulas for angular frequency?
ω=2πf
ω=2π/T
describe simple harmonic motion and give the defining equation.
simple harmonic motion is an oscillatory motion of an object where acceleration is always proportional to and opposite in direction to the displacement from equilibrium position.
a=-ω²x
recall an acceleration-displacement graph of SHM and describe how the graph shows SHM.
- straight line passing through origin shows acceleration is proportional to displacement from eqm position
- negative gradient indicates acceleration is opposite in direction to displacement from eqm position
of an object in SHM, describe:
1. its displacement from eqm position,
2. its velocity,
3. its acceleration,
4. its restoring (net) force,
at the eqm position and at the far ends of the oscillation.
at equilibrium:
x = 0
v = max
a = 0
Fnet = 0
at far ends:
x = max
v = 0
a = max, towards eqm position
Fnet = max, towards eqm position
what is the formula for max v and max a?
max v = (max x)ω
max a = (max x)ω²
against time, how to deduce equations for x, v and a?
the equations depend on the initial condition (when t=0, x=?) and sign convention given. if not specified, use simplest case found in data page.
for example, if x=0 at t=0, its a sin graph
starts in -x direction means -sin graph
for energy-time graphs, what are the equations for KE, PE and TE?
KE = ½mv² = ½m(x∘ωcosωt)²
TE = max KE = max PE = ½mx∘²ω² (same as energy-time)
PE = TE - KE = ½mω²(x∘sinωt)²
for energy-displacement graph, what are the equations for KE, PE and TE?
KE = ½mv² = ½mω²(x∘²-x²)
TE = max KE = max PE = ½mx∘²ω² (same as energy-time)
PE = TE - KE = ½mx²ω²
explain the term damping.
damping refers to the loss of energy from an oscillating system to the environment, due to dissipative forces, causing its amplitude to decrease with time.
what is the effect of light damping on amplitude and period?
decreasing amplitude exponentially with time (in both x and -x directions)
period increases slightly (no need show in graph)
what is the effect of no damping?
oscillation continues forever without coming to rest
amplitude and total energy is constant
what is the effect of heavy damping?
damping is so great that the object never oscillates but returns to its eqm position very very slowly (happens in very viscous liquid for example)
what is the effect of critical damping?
damping is just sufficient for time taken for the displacement to return to zero is a minimum.
describe a situation where critical damping is important.
shock absorbers in a car suspension system are designed to critically damp the suspension of the car so that passengers will not experience uncomfortable vibrations after the car passes over a hump.
what is meant by resonance?
it is a phenomenon whereby amplitude of a system undergoing forced oscillations is at a maximum.
occurs when frequency of periodic driving force is equal to the natural frequency of the system.
what are forced oscillations?
an oscillating system undergoes forced oscillations if
1) it is subjected to an external periodic driving force
2) as a result the frequency of the forced oscillations will be at the frequency of the driving force
what affects frequency response (amplitude)?
- magnitude of driving force
- degree of damping
- how close driving frequency is with respect to natural frequency of the driven system
