chapter 17 - oscillations Flashcards
shm demo spring and mass system
motion sensor connected to data logger and computer - under a spring with a mass attached oscillating
motion sensor - signal of known speed emitted then detected, time taken for echo to return is measured, distance = speed*time - must be in plane of motion, creates a movement plot
displacement (x)
distance from equilibrium position (m)
amplitude (A)
maximum displacement from equilibrium position (m)
period (T)
time taken for one full oscillation (s)
frequency (f)
no. oscillations per unit time (Hz)
angular frequency (⍵)
2𝞹/T = 2𝞹f
rad/sec
definition of shm
- acceleration is proportional to displacement
- acceleration acts in the opposite direction to the displacement
(restoring force always acting towards equillibrium)
a ∝ -x
a = -⍵x
isochronous
time period is independent of amplitude
- bigger A = bigger acceleration so faster over bigger distance = same T
equations of shm
⍵ = 2𝞹/T = 2𝞹f
x = Acos⍵t or Asin⍵t - cos if start at max displacement or sin is start at equilibrium
v = +- ⍵√A² - x²
vmax = +- ⍵A
a = -⍵²x
energy of shm
total energy is constant
PE goes from max to min at equilibrium to max
KE goes from min to max at equilibrium to min
EPE = 1/2kx²
KE = 1/2k(A²-x²)
damping
has the effect of reducing the amplitude of oscillations by applying an external force
damping examples
- air resistance bringing a mass-spring system to a stop
- shock absorbers in bike breaks increasing time to stop
- fire doors gave dampers so they shut at a controlled rate
3 types of damping
- light damping
- heavy damping
- critical damping
light damping
- period remains unchanged (isochronous)
- amplitude gradually decreases with time
heavy damping
- same as light but amplitude decreases more quickly
critical damping
no oscillations occur
free oscillations
- when a system is displaced from equilibrium position and oscillates without external forces
- this oscillation frequency is known as natural frequency
eg pendulum on string
forced oscillations
- when a system is driven by a periodic driving force
- the frequency of that driving force is known as the driving frequency
eg child moving legs on a swing
bartons pendulum experiment
penulum A provides a driving frequency that matches the natural frequency of pendulum B - as they are the same length
therefore moves with greater amplitude
real life examples of free and forced oscillations
- microwaves - microwaves (driving), water molecules (natural)
- MRI scanners - magnetic field (driving), water molecules in body (natural)
- ground resonance helicopters - rotor blade (driving), ground (natural)
resonance
when the driving frequency matches the natural frequency a system will resonate
causing amplitude of oscillations to increase
- undamped system will increase in amplitude as freq increase until natural (max) then decreases
- damped system will do the same but with a lower peak and moved left
amplitude that decays exponentiallys
if measured at the same time interval then the ratio between the displacement at those intervals is constant
resonance eg
eg paper on a mass spring system will have a lower natural freq and lower resonance peak then a normal mass spring system
shm graphs
start with displacement
acceleration is opposite displacement
velocity is the same as displacement but shifted to start at the opposite eg peak if displacement started at eq or eq is displacment started at peak
resonance and energy transfer
max energy transfer between driver and other
when driving freq is same as natural freq
when does a damped oscillator dissipate max energy
at t= 0 as has max speed so greatest friction
