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I need a little help completing a mechanics problem. If you have a pendulum put vertically up with a rigid rod and ball on top of the rod and you give the ball some initial velocity $v_x$ how would I find how long it would take the ball to hit the bottom of the swing (travel $\theta=180$)? I'm stuck.

Ok so I used energy to get: $$v=\sqrt{\frac{2}{m}(\frac{1}{2}mv_i^2+2mgL)}-2L(\cos\theta+1)$$ I am currently trying to find theta as a function of time so that I can integrate with respect to time.

It just dawned on me that I can't find theta as a function of time because it is a non-linear diff eq. So the only way I will be able to solve this problem is if I approximate (most likely with small angle approx.) which will throw my answer way off.

Shrodinger 2016
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    Perhaps the work in this answer is helpful? – Floris Aug 09 '16 at 03:39
  • The question is about a specific physics concept and shows some effort to work through the problem. Closing it is against the rules. – peterh Aug 09 '16 at 21:09
  • @peterh: What is the specific physics concept here? The question shows work and is thus better than most homework-like questions we get, but it still just asks how to solve a given exercise. – ACuriousMind Aug 10 '16 at 11:00

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Try thinking about some of the following:

1) Conservation of energy and how you can exploit $m*g*h$ to find the bob's velocity. Can you ignore the mass of the rod? That's a critical question.

2) The bob's velocity will always be tangential to the circular curve it's following.

3) Perhaps the relation $dS=R*d\theta$ can help when considering the circular path.

4) Does $dT=\frac{dS}{V}$?

5) Isn't $V$ dependent upon $\theta$ as it increases from $0$ to $180$?

Inquisitive
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  • Ok I was able to get velocity as a function of the angle using energy. If I can find the angle as a function of time I can get the time it takes by integrating. – Shrodinger 2016 Aug 08 '16 at 23:37
  • @Shrodinger2016 Try thinking about hints #2, #3, #4, and #5. Keep at it. It's not easy. – Inquisitive Aug 08 '16 at 23:42