If the horizontal displacement of a pendulum can be described by a cosine function, what is the precise equation for the displacement if the maximum swing is 15cm and a swing from right to left takes 2 seconds?
The displacement of a pendulum from rest can be represented by a cosine curve. The pendulum starts from a position 15 cm to the right of its rest position and swings 15 cm to the left of the rest position in 2 seconds. If, when you sketch the graph of the pendulum’s path, the vertical axis represents the displacement from the rest position and the horizontal axis represents time, write the function representing the motion.
The maximum swing either side of the rest position is +/- 15cm where a positive swing is to the right and a negative swing is to the left. So we want an amplitude of 15cm.
The pendulum takes 2 seconds to swing from the left to the right which is half a phase (a full phase is a swing to the left and then back to the right).
A full phase on the cosine curve is `2pi` on the x-axis, so a half phase is
Hence, to get the correct phase length we equate `t = (2x)/pi` where `t` is the time in seconds and `x` is the number of `pi`-radians on a standard cosine graph (when ` ``x=pi`, `t=2`).
Now `f(x) = cosx`
So `f(t)prop cos(pi/2 t)`
We wan` `t an amplitude of 15cm so
`f(t) = 15cos(pi/2t)` answer