1. The problem statement, all variables and given/known data Use the E-L equation to calculate the period of oscillation of a simple pendulum of length l and bob mass m in the small angle approximation. Assume now that the pendulum support is accelerated in the vertical direction at a rate a, ﬁnd the period of oscillation. For what value of a the pendulum does not oscillate? Comment on this result. 2. Relevant equations 3. The attempt at a solution I've got the first bit: L=(m/2)(l^2)(dθ/dt)^2-mgl(1-cosθ) E.O.M.: d2θ/dt2+(g/l)sinθ=0 d2θ/dt2+(g/l)θ=0 in the small angle approximation, which is S.H.M. with ω^2=√(g/l) (though I'm not sure about this as there's no minus sign in the E.O.M.) so T=2pi√(l/g) For the next bit, I just need help setting up the equations: So the generalized coordinates are θ and a. Are the following correct?: x=lsinθ y=-lcosθ+at (taking the origin as the point from which the pendulum is swinging)
Careful here, [itex]\omega[/itex] is that angular velocity in radians per time unit. Your T is the time to travel one radian (of the oscillatory cycle) not time per cycle. You need to multiply by [itex]2 \pi[/itex].
Yes. No. "a" is not a coordinate. Actually only θ is a generalized coordinate since the y coordinate of the support is constrained to be y=0.5at^2. The angle θ is the only freedom that the system has.