1. The problem statement, all variables and given/known data Hi, I have the following question... A pendulum consisting of a mass of 1kg is suspended from a string of length L. Air resistance causes a damping force of bv where b = 10-3 N/m 1) Derive and solve the equation of motion 2) Calculate the fractional decrease in amplitude of the pendulum oscillations if the pendulum is operated for 2 hours. 2. Relevant equations F=ma F=-kx x(t)=A0eαt 3. The attempt at a solution I think I have the derivation handled... F=ma=-kx-bv ma+bv+kx=0 m(d2x/dt2)+b(dx/dt)+kx=0 d2x/dt2+(b/m)(dx/dt)+(k/m)x=0 As this is a pendulum, I know x=Lsinθ and for small θ sinθ≈θ so... d2θ/dt2+(b/m)(dθ/dt)+(k/m)θ=0 I also know that k=mg/L so... d2θ/dt2+(b/m)(dθ/dt)+(g/L)θ=0 Therefore, I get the equation of motion for the dampened pendulum to be: d2θ/dt2+(b/m)(dθ/dt)+(g/L)θ=0 Now to solve the equation, I can turn it into a quadratic: α=(-(b/m)±(√(b2/m2)-(4g/L))/2 I can tidy this up a little, so... α=-(b/2m)±(√(b2/2m2)-(4g/L)) I know that √g/L = ω so and -√g/L = iω which I can take out as a factor so and tidy up the fraction inside the radical, so... α=-(b/2m)±iω√1-(b2L/2) I also know that I can let ϑ=ω√1-(b2/4m2ω2) so... α=-(b/2m)±iϑ I am assuming that this is the equation solved as I don't have a value for L and so can't go any further (can I?) As for calculating the fractional decrease, if I sub this back into the equation: x(t)=A0eαt I get... x(t)=A0e(-b/2m)t+eiϑt But I really don't know where to go from here, if, if anywhere? Do I simply rearrange the first part so.... x(t)/A0=e-18/5 so... x(t)/A0=0.0273 Any help gratefully received!