Solve Linked Masspoints Equation of Motion: α,l,m,g

[carllacan]

Homework Statement

Two particles of equal mass, one restricted to move along the y-axis and one restricted to move along the x axis, are linked by a solid rod of length l. Obtain the Lagrangian for the generalized coordinate α, defined as the angle of the rod with the horitzontal (see picture) and solve the equations of motion.
https://www.dropbox.com/s/h43s1749z7852oh/2014-01-30 22.50.54.jpg

Homework Equations

The Attempt at a Solution

My lagrangian looks like this: L = (m/2)l²\dot{α}² -mglsin(α)

From it I obtain \ddot{α} = (g/l) cos(α)

But I don't know how to solve that. I know how to use the small angle approximation for the case when there is a sin. Here I am clueless.

And I am algo given a hint which confounds me: I am told to solve for t(α) applying the following equation: https://www.dropbox.com/s/hl2xfm07wt40kdn/2014-01-30 23.00.23.jpg

 

[ehild]

Let \omega=\dot \alpha.

\ddot \alpha = \frac{d \omega}{d \alpha} \frac {d \alpha }{dt}=0.5\frac{d(\omega^2)}{d \alpha} =\frac {g}{l}\cos(\alpha)

Integrate. You get ω=dα/dt as function of alpha. Integrate again.

ehild

 

[carllacan]

Thank you, that was a clever trick! Now i can get to the equation mentioned on the hints. However I cannot go on, as the integral seems rather complex (i solved it with wolfram and ran out of computing time).

The hint actually tells me to give the result as t(α) applying the function F there defined. Would you interpret that as that I can just give t(α) = (some constants)*F(b, α)? (Yes, I know I should just ask the professor, but I have a few problems with that)

 

[ehild]

The hint actually tells me to give the result as t(α) applying the function F there defined. Would you interpret that as that I can just give t(α) = (some constants)*F(b, α)? (Yes, I know I should just ask the professor, but I have a few problems with that)

I would say you can use the hint given.


ehild