## Physics Friday 111

Consider a simple pendulum, with a mass m on the end of a rod of length l. In terms of the angle θ from the downward direction, the equation of motion for this pendulum is .

Now, suppose the pivot to which this pendulum is attached moves vertically in a sinusoidal motion of small amplitude, so that its vertical position is given by , with . What then is the new equation of motion? What happens for small-angle oscillations ()?

The position of the end mass in the plane of the pendulum, relative to the mid-cycle position of the pivot (where ), is .
Taking the time derivatives of both components,

,
and we can sum their squares to obtain the velocity squared of the mass:
,
and so the kinetic energy is thus
.
Similarly, our potential energy is simply , and so the Lagrangian is
.
and thus the Euler-Lagrange equation gives us equation of motion
.
Since , then , and so our equation of motion becomes
;
the gravitational acceleration g is replaced by . If we consider the (accelerating, non-inertial) reference frame of the pivot, the acceleration is , producing a proportional fictitious force, so that the net effect is the equation of motion above.

For small θ, then , and so our equation of motion becomes , or, dividing by l,
. The undriven pendulum has small-angle frequency ; and let us define . Using this, our equation of motion is
.
This equation cannot be solved analytically.