What Is the Maximum Angular Displacement of a Bob on a Frictionless Rail?

[sunnypic143]

Homework Statement

On a fixed frictionless horizontal rail is a massless ring to which is attached a bob of mass 'm' hanging vertically downwards with an inelastic and inextensible string. Starting from rest the ring is given a constant a constant horizontal acceleration (g/$$\sqrt{3}$$). What is the maximum angular displacement of the bob from the vertical?(Assume string is taut at all times)

Homework Equations

Well I have no serious idea on how to proceed. I tried force balance(weight, centripetal, tension) to get answer 30 degrees. But this may well be the equilibrium position and the bob may be expected to perform SHM so maybe this isn't the value asked for.(As I was intuitively under the impression that the bob will get displaced to a maximum under the influence of inertial forces and stay there without coming back; in a frame attached to the ring ; of course.).Was my intuition wrong?(If two people standing on either side of a vertical rod pivoted at the centre pull the opposite ends of the rod towards themselves with constant horizontal force, my intuition is that that the rod comes to rest at horizontal position, while it may be contended that the rod continues to rotate beyond its horizontal position as it had some angular velocity at its horizontal position.)
Assuming it was wrong, next I tried writing equations for vertical motion of the bob keeping track of the fact that acceleration of ring and bob along the string is equal at any instant.(Since the string is taut and inextensible.)
Please give me complete guidance on both the above approaches along with any simpler ones if possible.(perhaps energy and momentum methods)

 

[tiny-tim]

Hi sunnypic143!

My inclination would be to use the principle of equivalence … the horizontal acceleration of the ring is equivalent to a horizontal gravitational acceleration in the (accelerating) frame in which the ring is staionary.

Add that to the real gravitational acceleration to get the total gravitational acceleration, at an angle θ, say, to the vertical.

Then just treat the pendulum as if it was swinging about this new "vertical" angle.