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Pendulums
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.
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.
, 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.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 
.
. Conversely, the kinetic energy and velocity
of the pendulum are maximized at the equilibrium point and minimized
when . The figure below summarizes this information 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.
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
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:
. 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.
). It’s highly unlikely
that you’ll be asked to give a specific number.|
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