(adsbygoogle = window.adsbygoogle || []).push({}); This is the problem (exact wording):

An infinitely long rod is being rotated in a vertical plane at a constant angular velocity w about a fixed horizontal axis (the z-axis) passing through the origin. The angular velocity is maintained at the value w for all times by an external agent. At t = 0 the rod passes through zero-inclination, i.e., q = 0 at t = 0 where q is the angle the rod makes with the x-axis. There is a mass m on the rod. The mass' coordinates and velocity components at t=0 are

[tex] r(0) = \frac{g}{2 \omega^2} [/tex]

[tex] \theta(0) = 0 [/tex]

[tex] \dot{r}(0) = 0 [/tex]

[tex] \dot{\theta}(0) = \omega [/tex]

where g is the acceleration due to gravity. The mass m is free to slide along the rod. Neglect friction. Hint: Recall that in plane polar coordinates the unit vectors and are not constant.

http://electron6.phys.utk.edu/phys594/archives/mechanics/images/1997m/Image509.gif [Broken]

(a) Find an expression for r(t), the radial coordinate of the mass, which holds as long as the mass remains on the rod.

This is what I tried:

Let T be the kinetic energy of the 'system', so that in polar coordinates,

[tex] T = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) = \frac{1}{2}m(\dot{r}^2 + r^2\omega^2) [/tex]. (I wasn't given the expression for the speed, so had to derive it from first principles, so it is right?)

Let U be the potential energy of the system, so that:

[tex] U = mgrsin\theta = mgrsin(\omega t) [/tex]

Therefore, the Lagranian is:

L = T - U, so: [tex] L = \frac{1}{2}m(\dot{r}^2 + r^2\omega^2) - mgrsin(\omega t) [/tex]

[tex] \frac{d}{dt} ( \frac{\partial L }{\partial \dot{r}} ) = \frac{\partial L}{\partial r} [/tex], so:

[tex] m \ddot{r} = mr\omega^2 - mgsin(\omega t) [/tex]

My last question is this: To solve the problem, do I need to solve this differential equation? If so, how? - I've had an attempt, but I'm quite sure that my answer is wrong. I won't post it on here unless someone wants to see it. I've never actually solved a second order differential equation before (first order, yes). Thanks in advance.

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# Lagrangian Problem that I'm stuck on.

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