Solving Emergency Problem Involving Rod Rotating Around Z Axis

[ShayanJ]

There is a rod rotating around the z axis with angular velocity $\omega$.The angle between the rod and z axis is constant and equal to $\alpha$.A mass m is constrained to move on the rod.The gravitational force is in the negative direction of z axis and the friction between the mass and rod is negligible.At time t=0 , the mass is at distance $r_{0}$ from the origin and is stationary relative to rod.

1-In a proper coordinate system,write the Newton's second law for the mass in a inertial frame of reference and write the differential equations of motion.Highlight the expressions indicating the reaction of rod.

2-Solve the differential equations and write the mass's distance from origin as a function of time.

3-Calculate the reaction force of the rod(direction and magnitude)


thanks

 

[Spinnor]

Does the following get you started?

 

[ShayanJ]

Thanks a lot.My mistake was that I didn't write N but only the centripetal force.

But it seems sth is wrong there.Just imagine such a thing not rotating.you know that the mass slides down.So it seems that $N \sin {\alpha}$ is not equal to mg.
Another question.
Last night I worked on it and got some things but there is a big puzzle in my mind.
Textbooks say that the acceleration in spherical coordinates is as following:

$\textbf{a}=(\ddot{r} - r \dot{\phi}^{2} \sin ^ {2} \theta -r \dot{\theta}^{2}) \hat{e_{r}} +$ $( r \ddot{\theta} + 2 \dot{r} \dot{\theta} - r \dot{\phi} ^ {2} \sin {\theta} \cos{\theta}) \hat{e_{\theta}} +$ $(r \ddot{\phi} \sin{\theta} + 2 \dot {r} \dot {\phi} \sin {\theta} + 2 r \dot{\theta} \dot{\phi} \cos{\theta}) \hat{e_{\phi}}$

Shoud I use the r coordinate of the above equation for writing Newton's 2nd law or simply write $m \ddot{r}$ ?
I tried both.When I use the equation above I get crazy things.But when I use $m \ddot{r}$ I get a cosine.
I'm just wondering that the above equation is the most general and can't understand why it becomes wrong in this problem.

The last question.
The $\theta$ coordinate of the acceleration should be zero.But when I write the Newton's 2nd law in that coordinate and replace $\ddot{\theta}$ with zero,I get r=constant,which is clearly wrong.Could you write that?

thanks

 

[pymn_nzr]

Do you know what i said?