A pendulum is defined as a mass, or bob, connected to a rod or rope, that experiences simple harmonic motion as it swings back and forth without friction. The equilibrium position of the pendulum is the position when the mass is hanging directly downward.
Consider a pendulum bob connected to a massless rope or rod that is held at an angle
from the horizontal. If you release the mass, then the system will swing to position
and back again.
The oscillation of a pendulum is much like that of a mass on a spring. However, there are significant differences, and many a student has been tripped up by trying to apply the principles of a spring’s motion to pendulum motion.
Properties of Pendulum Motion
As with springs, there are a number of properties of pendulum motion that you might be tested on, from frequency and period to kinetic and potential energy. Let’s apply our three-step method of approaching special problems in mechanics and then look at the formulas for some of those properties:
- Ask yourself how the system will move: It doesn’t take a rocket scientist to surmise that when you release the pendulum bob it will accelerate toward the equilibrium position. As it passes through the equilibrium position, it will slow down until it reaches position
, and then accelerate back. At any given moment, the velocity of the pendulum bob will be perpendicular to the rope. The pendulum’s trajectory describes an arc of a circle, where the rope is a radius of the circle and the bob’s velocity is a line tangent to the circle.
- Choose a coordinate system: We want to calculate the forces acting on the pendulum at any given point in its trajectory. It will be most convenient to choose a y-axis that runs parallel to the rope. The x-axis then runs parallel to the instantaneous velocity of the bob so that, at any given moment, the bob is moving along the x-axis.
- Draw free-body diagrams: Two forces act on the bob: the force of gravity, F = mg, pulling the bob straight downward and the tension of the rope,
, pulling the bob upward along the y-axis. The gravitational force can be broken down into an x-component, mg sin
, and a y-component, mg cos
. The y component balances out the force of tension—the pendulum bob doesn’t accelerate along the y-axis—so the tension in the rope must also be mg cos
. Therefore, the tension force is maximum for the equilibrium position and decreases with 
. The restoring force is mg sin 
, so, as we might expect, the restoring force is greatest at the endpoints of the oscillation, 
and is zero when the pendulum passes through its equilibrium position.
You’ll notice that the restoring force for the pendulum,
mg sin
, is not directly proportional to the displacement of the pendulum bob,
, which makes calculating the various properties of the pendulum very difficult. Fortunately, pendulums usually only oscillate at small angles, where sin
. In such cases, we can derive more straightforward formulas, which are admittedly only approximations. However, they’re good enough for the purposes of SAT II Physics.
