Simple Harmonic Motion Flashcards
describe the forces in a string when pendulum is at rest position
- resultant force is zero
- tension is upwards and equal to the weight
describe the resultant force when pendulum is oscillating
always acts towards equilibrium position (rest position)
describe the conditions needed for SHM
- acceleration is proportional to displacement from fixed point*
- direction of acceleration always towards fixed point
*cannot use newtons laws as a varies with s
what does the minus sign in the acceleration equation mean
displacement and acceleration are in opposite directions
force equation
F = ma
∴
F = -mω²𝑥
displacement-time graph
𝑥 = A cos ωt and +ve
∴
starts at max displacement and top half
top sin, bottom cos
velocity-time graph
v = -ωA sin ωt and -ve
∴
starts at zero and goes below
top -sin, bottom cos
acceleration-time graph
a = -ω²𝑥 = -ω²A cos ωt
∴
starts at max and down below
top -cos, bottom -sin
velocity and displacement eqaution
v = ±ω√(A²-𝑥²)
± indicates direction of movement (can eb left/right/up/down)
displacement equations
- variation with time
- max
- min
- 𝑥 = A cos ωt
- equal to amplitude
- 0
velocity equations
- variation with time
- max
- min
- variation with displacement
- v = -ωA sin ωt
- ±ωA
- (extreme displacement) =0
- a = -ω²𝑥
acceleration equations
- variation with time
- max
- min
- variation with displacement
- a = -ω²A cos ωt
- (extreme displacement) = ω²A
- 0
- a = -ω²𝑥
total energy with time
constant
kinetic energy at an instant during SHM
KE = ½mv²
* find v by gradient of displacement-time graph
OR
* substitute v = ω²A sin ωt
time period for simple pendulum equation
T = 2π√l÷g
time period for a loaded spiral spring equation
T = 2π√m÷k
undamped oscillations
- free vibrations (perfect SHM)
- displacement caries with time
- amplitude constant with time
lightly damped
- displacement varies with time
- amplitude gradually gets smaller until oscillations stop
over-damped
- doesn’t oscillate when displaced
- reutrns slowly to equilibrium
critical damping
- system returns to equilibrium in shortests possible time (¼T)
what is a forced vibration
when any external froce that varies with time is used to make an object oscillate
when does resonance take place
when the frequency of the driving force is the same as the frequency of the oscillating system (object with same length as driving force has largest amplitude)
examples of resonance
- destructive in mechanical systems
- to make sound more audible in musical instruments
- producing sound that is barely heard with tuning forks
- electrical circuits in radio and receivers
- microwaves
