Classical Mechanics

Classical Mechanics

1. Conservative Forces and Energy

   • T=⅟₂mẋ
   • W= ∫F dx =⅟₂ mẋ²(t)-⅟₂mẋ²(t)
   • E= ⅟₂mẋ²+V(x(t))

2. Stability and Equilibrium

   • E≥V(x) is BOUNDED MOTION
   • E=V(x), ẋ=0, PARTICLE IS STATIONARY
   • EQUILIBRIUM POINT: 0=dv/dx (x)

3. Second Order Equations

   • If D>0. λ₁, λ₂ exist.
     • Gen Sol: x(t)= C1e^λ1t + C2e^λ2t
   • If D<0 z1,2 exist as complex conjugate
     • Gen Sol: x(t)= e^ λt(C1cos(ϴt)+C2sin( ϴt))
   • If D=0, z1=z2
     • Gen Sol: x(t)= (C1+C2)e^λt

4. Imulsive Motion

   • Coefficient of restitution, e
     • e=o, coalesce
     • 0<e<1, |v1-v2|=e|u1-u2|
     • e=1, elastic coll. No los in KE
     • e>1, Super elastic.
   • Linear Momentum
     • p(t1)-p(t0)= ∫ F dt
     • p=m. d/dx (r)
   • Momentum is p=mV