Bead sliding along rotating rod
The setup is given in Fig1. A rod is connected with evolute joint to stable support S. A torque T rotates the rod about O. Its value is arranged such that rod has a constant angular acceleration α. A bead, having mass M, is free to slide, without friction, along the rod.
![[Graphics:HTMLFiles/bead_sliding_free_on_rotating_rod_4_1.gif]](HTMLFiles/bead_sliding_free_on_rotating_rod_4_1.gif)
The governing differential equations can be derived as,
![M r^′′[t] = M α^2 t^2 r[t] ... ... (1)](HTMLFiles/bead_sliding_free_on_rotating_rod_4_2.gif){kind=small}
![M ( 2 α t r^′[t] + α r[t] ) = F_t ... ... (2)](HTMLFiles/bead_sliding_free_on_rotating_rod_4_3.gif){kind=small}
It follows immediately from (1) that a force is acting along the rod, directed away from the center. In effect, we have here a real centrifugal force acting on the particle, giving the bead its radial outward acceleration r''[t].
![Centrifugal Force = M α^2 t^2 r[t] ... ... (3)](HTMLFiles/bead_sliding_free_on_rotating_rod_4_4.gif)
The transverse inertial forces acting on the bead are given by (2).
The solution derived for differential equation (1) is:
![r[t] = ^(-0.5 t^2 α) ( Hypergeometric1F1[0.25, 1/2, t^2 α]) ... ... (4)](HTMLFiles/bead_sliding_free_on_rotating_rod_4_5.gif)
Using the solution (4) above one can derive the trajectory of the bead as shown in Fig2.
![[Graphics:HTMLFiles/bead_sliding_free_on_rotating_rod_4_6.gif]](HTMLFiles/bead_sliding_free_on_rotating_rod_4_6.gif)
Fig2
Conclusion
We have shown mathematically a real centrifugal force to act on the bead, causing the outward acceleration of the bead along the rod. This centrifugal force is also readily measured with a micro wireless accelerometer attached to the bead. It is interesting to note that the centrifugal force in this particular case is acting without the usual presence of a centripetal force.
mandrin
