2 - 11 - applications of differential eq Flashcards
setting up differential eq
use F = ma and a = dv/dt
shm differential eq
d^2x/dt^2 = -⍵^2 x
shm general solution
m^2 = -⍵^2
m = +-⍵
x = Asin⍵t + Bcos⍵t
x = Rsin(⍵t + Φ)
equilibrium position
average position around which the object oscillates
amplitude
max displacement from equilibrium
period
time after which the motion repeats
= 2pi / ⍵
angular frequency
⍵
if initially at eq vs displacement solution
equilibrium - x = a sin⍵t
max displacement - x = a cos⍵t
velocity
v^2 = ⍵^2(a^2 -x^2)
max v = ⍵a
drag force
D = -kv
differential eq shm damped
d^2x/dt^2 + k dx/dt + ⍵^2 x = 0
differential eq shm damped solutions
k^2 -4⍵^2 >0
- 2 real roots
- x = Ae^m1t + Be^m2t
- overdamping
k^2 -4⍵^2 =0
- repeated roots
- x = (A + Bt) e^-mt
- critical damping
k^2 -4⍵^2 <0
- complex roots
x = e^-pt (Asinqt + Bcosqt)
- under damping
over damping
k^2 -4⍵^2 >0
- 2 real roots
- x = Ae^m1t + Be^m2t
critical damping
k^2 -4⍵^2 =0
- repeated roots
- x = (A + Bt) e^-mt
under damping
k^2 -4⍵^2 <0
- complex roots
x = e^-pt (Asinqt + Bcosqt)
linear systems
- get dy/dt and dx/dt in terms of y and x
- differentiate dy/dt with respect to t
- sub in dx/dt
- rearrange the first eq in terms of x= and sub in
- left with an equation with only y
- solve like a normal differential eq
