We return again to our driven pendulum. This time, we consider what happens when the pendulum is near the upright vertical position, *θ*=π, the unstable equilibrium of the undriven pendulum. To do this, let us denote a new angle measure , so that measures the angle from the upright vertical. Then , , and in terms of , the equation of motion

becomes

;

and for small , this is approximated by

.

Now, one might expect that the pendulum will fall over, as it does in the undriven case. After all, the tangential acceleration, , appears like it should average over a cycle to , a positive quantity. However, this reasoning is incorrect in one important way: recall the “wiggle” in *θ*, and thus in , due to the driving, and also proportional to a . By examination, we see that when the pivot is accelerating upward, will be driven to increase, and a downward acceleration will produce the part of the wiggle decreasing (or slowing the increase in) .

Thus, the in the driving term of the acceleration will contain a cosine term, so the above will have a cosine squared term, which, like in our previous rapid *ω*, small *α* approximation, will have a nonzero average. In fact, we see that due to the wiggle, will be slightly larger when the driving acceleration is towards lower , and thus this will have a greater effect than when the driving is in the opposite direction, and is slightly smaller. Thus, if *ω* is large enough, then the net effect of the driving is a restoring force stronger than gravity, and one gets the pendulum oscillating about the upright vertical, with a small wiggle superimposed!

Working as we did in our approximation here, we try for an approximate solution of the form , where *b*(*t*) and *c*(*t*) are functions whose variation over time is much slower that the “wiggle”, so that on a time-scale of , with , we can ignore their time dependence and treat them like constants; we also expect *b* to be much smaller than *c* (as it represents the “wiggle”).

As before, we substitute into our equation of motion:

.

Now, with , and rapid enough driving that , we can, as before, ignore the term to get

.

Which, to leading order, tels us that , so , and so our approximate solution is

.

From our equation of motion, we have acceleration of *φ* given by

.

Now, taking the average over a period of the driving (), we see that in the product , we will have four terms; the constant term will obviously average to itself, while the two cosine terms will each average to zero, and since the average of over one of its periods is 1/2, the fourth term, will have average . Lastly, *c*(*t*) is approximately constant on this timescale.

Therefore,

.

Suppose that , so that the quantity in brackets is positive. Assuming this is true, then define . Then

.

But from , we see that on the timescale we are averaging, , and so we have

,

which is (in the approximation) simple harmonic motion of frequency

.

Now, for this to happen, we noted that must be real. With , we see then that to have , we need , as was assumed earlier.

Note that if (instead of just ), then .

Thus, in this approximation, our pendulum swings with a frequency ; with a small wiggle of frequency superimposed. We also note that the amplitude of the “wiggle” is proportional to the current angle about which it is wiggling, being smaller by a factor of , creating behavior similar to that here, except with a distinctly different , and with a wiggle of opposite phase.

Tags: Approximation, Friday Physics, Pendulum, physics

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