# Quick Lagrangian of a pendulum question

1. ### Lucy Yeats

117
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,

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)

Last edited: Mar 5, 2012
2. ### jambaugh

1,802
Careful here, $\omega$ 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 $2 \pi$.

3. ### Lucy Yeats

117
Thanks for pointing that out, I'll correct that in the first post. :-)

4. ### Lucy Yeats

117
Any help would be great. :D

117

6. ### I like Serena

6,194
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.

7. ### Lucy Yeats

117
Great, I've got it now.

Thanks! :-)