17 - Oscillations Flashcards
simple harmonic motion is an oscillating motion where the…
acceleration is proportional to displacement
acceleration (and resultant force) always in opposite direction to displacement - towards equilibrium position
a∝-w^2x
x=Asinwt
v=dx/dt = wAcoswt
max V when coswt = 1
max v =wA
max v = 2πfA
a=-w^2x , what is max a
max a when x is max amplitude
so a max = -w^2A
resonance is
amplitude of oscillation increases as driving frequency of external force applied is same as natural frequency of object
object oscillates at driving force frequency
(if undamped, amplitude will continue to increase until system fails)
at resonance =
natural frequency of system is equal to the applied frequency of the periodic force (in phase)
what does damping decrease (graph)
amplitude lower for all frequencies
max amplitude occurs at lower frequency than natural f
peak becomes flatter, broader, graph shifts to left
natural frequency
frequency of a system oscillating without external periodic force
periodic force
force with regularly changing amplitude and definite time period
forced oscillation
oscillation under external periodic force, causing it to oscillate at a particular frequency
when oscillation is at left
x = max (-)
v = 0
a = max (+)
F = max (+)
when oscillation is at midpoint/equilibrium
x = 0
v = max (+/-)
a = 0
F = 0
when oscillation is at right
x = max (+)
v = 0
a = max (-)
F = max (-)
Vmax
Vmax = 2πfA
a.max
a.max = -(2πf)^2 A
where x = a due to max displacement
acceleration and force
are linked and both act towards equilibrium, opposite direction to displacement
energy changes as mass spring system goes from lowest to highest point
KINETIC ENERGY - zero at lowest, increases to maximum at equilibrium then decreases to 0 at highest
GRAV POTENTIAL ENERGY - increases
ELASTIC POTENTIAL ENERGY - decreases
TOTAL ENERGY decreases due to damping effect of air, air gains thermal energy
total energy is equal to
maximum kinetic energy
the period of a simple harmonic oscillator…
is independent of its amplitude (isochronus oscillator - same time)
what is a free oscillation
when an object oscillates at its natural frequency, without any eternal forces applied
where is kinetic energy max
at equilibrium point, where velocity is a maximum
where is maximum potential energy max
occurs at amplitude positions (max displacement).
damping and energy
resistive forces reducing amplitude of oscillations
energy loss to resistive forces like drag and friction
kinetic energy of oscillations is transferred to other forms such as heat
types of damping
light - naturally like oscillating in air - amplitude decreases, period unchanged
heavy - oscillating in water, amplitude decreases significantly
critical - oscillating in treacle, object stops before one oscillation, moves to equilibrium, no oscillatory motion
use of damping
suspension forks on mountain bikes - damping uses to absorb energy of shock. if not, it would oscillate up and down
frequency of free oscillation
= natural frequency of oscillation
resonance and energy
at resonant frequency, greatest possible transfer of kinetic energy from driver to oscillator
damping may reduce at a rate that is…
exponential - amplitude decreases by same factor in equal time intervals
damping force is in what what direction to velocity
damping force is maximum when
opposite to velocity
max. when displacement is zero as speed is greatest so damping force is greatest in magnitude
example of forced oscillation
loudspeaker oscillating and producing a continuous note
Barton’s pendulum - paper cones diff lengths suspended along with heavy brass bob - driving force making all cones oscillate at same frequency. the cone w same length as brass bob has same natural f so will resonant and have greater amplitude
mass is hung from bottom of spring. explain how it can be made to show resonance
force mass to oscillate w/ periodic force. driving must equal natural frequency of mass-spring system, increasing amplitude of oscillation - resonance
when does an oscillator dissipate maximum energy
at equilibrium - max speed so max kinetic energy and greatest friction
experiment to obtain displacement/time graph for oscillating mass in lab
motion sensor connected to data logger to get info on displacement and time shown on connected computer
motion sensor placed close to mass to eliminate reflections from other objects
reflector should not cause damping
motion sensor should be directed along line of oscillation
example of resonance
microwave oven - drving freq - em radiation which matches the natural freq of water molecules causing resonance. amplitude increases and kinetic energy of molecules in food increases
child on swing
wind and bridges
