Define simple harmonic motion
the acceleration of an object is proportional to its displacement from a fixed point and the direction of acceleration is always towards that point
periodic time
time taken for one complete oscillation
what is simple harmonic motion an example of?
particular type of oscillation where period of oscillation is not affected by the amplitude of the oscillation
What are the 2 conditions that must be met for SHM to occur?
acceleration is always in the opposite direction to the displacement from a fixed point (the equilibrium position)
acceleration is proportional to the displacement from the fixed point
simple harmonic acceleration formula
a= -ω2x
where
a = acceleration
ω = angular frequency
x = displacement from fixed point
simple harmonic displacement formula
x = A cos ωt
where
x = displacement
A = amplitude
t = time
For the simple harmonic displacement formula to be valid, what must happen?
cos ωt must be calculated in radians
simple pendulum equation
T = 2π root l/g
where
t = periodic time
l = length of string
g = gravitational acceleration
What is the assumption made in the simple pendulum equation?
angle <15 for a small amplitude and small oscillations
Damping forces
resistive forces that act on an oscillating system
free oscillations
oscillations are those which are not damped
What are undamped oscillations?
free oscillations
What a system is lightly damped, what occurs to the amplitude and displacement?
displacement will vary with time as with free oscillations
amplitude of oscillation will gradually get smaller and eventually oscillations will cease
Over-damped system
no oscillations occur
system returns very slowly to equilibrium position
critical damping
system can return to its equilibrium position in shortest possible time
forced vibration
occurs when any external force which varies with time is used to make an object oscillate
When does resonance occur?
when the frequency of the driving force is the same as the natural frequency of the oscillating system
4 examples of resonance
air column in a resonance tube
oscillating loaded spiral spring
vibrating string of a musical instrument
water molecules in food when placed in microwave oven
What is the equilibrium position in SHM?
where object would naturally rest in absence of accelerating force
What two conditions must be in place for a mechanical system to be able to oscillate?
system must be able to store potential energy
system must have some mass, enabling it to have kinetic energy
What is the amplitude, A, of SHM?
magnitude of maximum displacement and always positive
For SHM, how is the velocity related to displacement?
velocity is at a maximum for zero displacement and is zero for maximum displacement
What shape is the displacement - time graph for a particle which begins SHM at maximum displacement?
cosine curve
What is meant by the natural frequency of an object?
frequency that an object will choose to oscillate if given the opportunity
