Period
The period of oscillation of the pendulum, T, is defined in terms of the acceleration due to gravity, g, and the length of the pendulum, L:
This is a pretty scary-looking equation, but there’s really only one thing you need to gather from it: the longer the pendulum rope, the longer it will take for the pendulum to oscillate back and forth. You should also note that the mass of the pendulum bob and the angle of displacement play no role in determining the period of oscillation.
Energy
The mechanical energy of the pendulum is a conserved quantity. The potential energy of the pendulum,
mgh, increases with the height of the bob; therefore the potential energy is minimized at the equilibrium point and is maximized at
. Conversely, the kinetic energy and velocity of the pendulum are maximized at the equilibrium point and minimized when
.
The figure below summarizes this in a qualitative manner, which is the manner in which you are most likely to find it on SAT II Physics. In this figure,
v signifies velocity,
signifies the restoring force,
signifies the tension in the pendulum string,
U signifies potential energy, and
KE signifies kinetic energy.
Velocity
Calculating the velocity of the pendulum bob at the equilibrium position requires that we arrange our coordinate system so that the height of the bob at the equilibrium position is zero. Then the total mechanical energy is equal to the kinetic energy at the equilibrium point where
U =
0. The total mechanical energy is also equal to the total potential energy at
where
KE =
0. Putting these equalities together, we get
But what is h?
From the figure, we see that
. If we plug that value into the equation above, we can solve for
v:
Don’t let a big equation frighten you. Just register what it conveys: the longer the string and the greater the angle, the faster the pendulum bob will move.
How This Knowledge Will Be Tested
Again, don’t worry too much about memorizing equations: most of the questions on pendulum motion will be qualitative. There may be a question asking you at what point the tension in the rope is greatest (at the equilibrium position) or where the bob’s potential energy is maximized (at
). It’s highly unlikely that you’ll be asked to give a specific number
