Lecture d31. Linear harmonic oscillator Spring-Mass system k n Spring force F k>0 Newton's second Law m2+k:x=0 (Define) Natural frequency(and period) k 2丌 Equation of a linear harmonic oscillator +wnx=o
Lecture D31 : Linear Harmonic Oscillator Spring-Mass System Spring Force F = −kx, k > 0 Newton’s Second Law mx¨ + kx = 0 (Define) Natural frequency (and period) ωn = s k m τ = 2π ωn Equation of a linear harmonic oscillator x¨ + ω 2 nx = 0 1
Solution General solution c(t=A cos wnt+B sin wnt or xc(t)=Csin(unt+φ) Initial conditions x(0)=x0x(0)=io Solution w(t=o cos wnt+-sin wnt or 1r0 a(t)=va5+(co/wn)sin(wnt +tan( ))
Solution General solution x(t) = A cos ωnt + B sin ωnt or, x(t) = C sin(ωnt + φ) Initial conditions x(0) = x0 x˙(0) = ˙x0 Solution, x(t) = x0 cos ωnt + x˙0 ωn sin ωnt or, x(t) = q x 2 0 + ( ˙x0/ωn) 2 sin(ωnt + tan−1 ( x0ωn x˙0 )) 2
Graphical Representation 2丌 T B t Displacement, velocity and Acceleration 0 2丌
Graphical Representation Displacement, Velocity and Acceleration 3
Energy Conservation st Equilibrium Position No dissipation T+V= constant otential Energy k(a+sst) k 7709 At Equilibrium -kost f mg=0 V=-kr 2
Energy Conservation Equilibrium Position No dissipation T + V = constant Potential Energy V = 1 2 k(x + δst) 2 − 1 2 kδ2 st − mgx At Equilibrium −kδst + mg = 0, V = 1 2 kx2 4
Energy Conservation(cont'd) Kinetic Energy Conservation of energy (T+V)=mii+ kai=0 Governing equation mi+k= o Above represents a very general way of de- riving equations of motion (Lagrangian Me- chanics
Energy Conservation (cont’d) Kinetic Energy 1 2 mx˙ 2 Conservation of energy d dt (T + V ) = mx˙x¨ + kxx˙ = 0 Governing equation mx¨ + kx = 0 Above represents a very general way of deriving equations of motion (Lagrangian Mechanics) 5