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}}<br />

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?