Pendulums

Written by tutor John Z.

If you have ever been on a swing, you have acted like a pendulum. A pendulum can be loosely defined as something hanging that sways from one extreme
to the other (oscillates). You have probably heard of grandfather clocks and oscillatory rides from amusement parks as other examples of pendulums.
A metronome is right side up but can be described as a pendulum because it swings back and forth on a rigid rod. The seasons can also be described as
oscillating from one extreme to the other. As you may see from many of these examples, pendulums are excellent timekeepers.

In the above graphic, L is the length of the string. Θ is how many degrees past the vertical the pendulum swings.

The equilibrium is the point where the pendulum would rest if it were not swinging. For a simple pendulum perpendicular this is always the middle,
provided the pendulum is perpendicular to the surface of the earth. There are always two extreme points which are an equal distance from the equilibrium
point. A simple pendulum will swing from one extreme point through the equilibrium position to the other extreme point. There, it will reverse direction,
swing through the equilibrium point, and repeat the process. The maximum angle that a pendulum swings, or its most extreme position, is defined as its
“amplitude (A).” The amount of time is takes to go from one extreme point to the other and back is defined at the “period (T).” The
“frequency (f)” is defined by how many cycles the system goes through per second. Frequency is related to the reciprocal of the period, so you
only need to know one of them. You may have heard these terms in waves or springs and it is not an accident that the same words are used. Pendulums,
spring-mass systems, waves, and orbits have a lot it common.

Why does a pendulum oscillate?

Well, first off, if a pendulum is just resting at the equilibrium position and no force is applied to it, it will continue to stay at rest, just like
Newton said it would. The magic happens when it is disturbed
from its equilibrium position. The moment it is moved from the middle, it will feel a “restoring force” pulling it back to the equilibrium position.
The further the pendulum is from its equilibrium position, the stronger the restoring force will be pulling it back. The restoring force is therefore
greatest at the extreme points and minimal (zero) at the equilibrium point. This restoring force causes an increase in velocity on the way back to the
equilibrium position and overshoots it. The velocity of the mass at the extreme points is minimal (zero) and the velocity at the equilibrium position
is maximum but more on that later. The moment the pendulum passes through the equilibrium position in the other direction, it feels a restoring force
pulling it back to the middle. This restoring force slowly reduces the velocity of the mass until it is 0 again when it reaches the opposite extreme point.
Then the velocity switches directions, accelerating the mass toward the equilibrium, and the process starts over.

	Restoring Force	Velocity
When at Equilibrium	0 (proportional to how far it is from equilibrium)	Maximum (from conservation of energy)
When at Amplitude	Maximum     which causes ->	0

In a completely idealized simple pendulum, there will be no energy losses and the pendulum will swing back and forth forever. For the
case of the swing, we would have to take into account many real world issues including the center of mass of the system (why do you
swing your legs forward and lean back?); however, for this discussion, we will only be concerned with simple pendulums, a ball hanging
from a string. We will assume we are in a frictionless world, with massless strings that always remain taut, no air resistance, a
perfectly uniform ball, and idealized in any other way we want for simplicity. Obviously, this cannot exist in the real world but it
is beneficial to be able to account for the greatest effectors to get a ball park estimate for a real life problem. It also makes it
easier to add in the smaller effects later.

What is the period of a pendulum?

Remember, the period of anything is the amount of time for one full revolution. The period (T), with SI units of seconds, of a simple
pendulum oscillating at small angles (less than 20°) is:

As you can see, the period for a simple pendulum with a small oscillatory angle is only dependent on the length (L) of the pendulum
and the local acceleration due to gravity (on the surface of Earth, g = 9.81 m/s). It is surprisingly independent of the mass on the
string. It is also even more surprisingly independent of its amplitude. A pendulum oscillating at 5 degrees will have roughly the same
period as one with 15 degrees if they have the same length and are on the same planet. This is what makes pendulums like grandfather
clocks and metronomes such great time keeping devices. Even as real world effects like friction or air resistance suck energy out of the
system, the period remains constant. Some clocks even compensate for this by increasing the length of the string as it loses energy.
Because frequency (f) of a pendulum is equal to 2π/T:

What is the restoring force for a simple pendulum?

As you can see in the above free body diagram of a pendulum that it not at its equilibrium position, the gravity vector can be broken
up into a component equal and opposite to the tension in the string and into a component perpendicular to it that points directly toward
the equilibrium position no matter what the displacement angle. When at the equilibrium position, gravity is opposite the tension in the
string with no other components. Remember that the restoring force is greatest when it is far away and zero when at equilibrium. Gravity
is the restoring force that continually attempts to pull pendulums back to their equilibrium position. Another way to look at it is since
the string of the simple pendulum is assumed to be taut or rigid, when the pendulum is at its equilibrium position, it is at its lowest
point. When it is anywhere else, it has more gravitational potential energy than at the equilibrium position. Since no energy is lost, we
can use the conservation of energy to relate the height and tangential velocity of the pendulum at any point through the definition of
gravitational potential energy and kinetic energy.

Conservation of Energy

Through an energy balance or by applying the conservation of energy
to a pendulum, we can relate the length of the pendulum to the tangential velocity through its max height at its amplitude.
The height at any point can be found through trigonometry to be the length of the pendulum as measured at the equilibrium position
minus the distance from the pivot point at its amplitude or

H = L – LcosΘ = L(1 – cosΘ)

At the pendulum’s highest point, it has all potential energy and no kinetic energy since its velocity is zero. When passing through its
equilibrium position, it has all kinetic energy and no gravitational potential energy since it is at its lowest point. We will measure
change in height from this lowest point and so we can say:

The same mass is being multiplied on both sides so it will cancel out showing that the tangential velocity is independent of the mass
of the object swinging. Solving for velocity yields:

Similarly,

We can use this information to solve to the change in height or tangential velocity at any point by doing a conservation of energy
knowing that the system has some amount of potential and kinetic energy but it is always equal to the amount of energy at equilibrium
and the extreme points, which are easily calculable. Again, mass will cancel out and will have no effect on the system.

Note that since the tangential velocity is not constant along the semicircular path, centripetal motion is not applicable for the entire
path. However, you can set centripetal force equal to tension while it is going through the equilibrium position; some strings may break
if the mass or the velocity of the ball is too big.

Motion of a pendulum:

The motion of a pendulum can be expressed as a function of displacement from the equilibrium postion (Θ) and time (t):

When you plug in 0 for time, you get a displacement angle of 0 because the pendulum is at the equilibrium position. When the function
is equal to zero, it is at its equilibrium position. When the function equals its amplitude, it is at its extreme position, or amplitude.
Because sin is a periodic function, it will cross zero an infinite amount of times and therefore the pendulum will cross the equilibrium
position an infinite amount of times. The trick is to figure out how often it does this. Remember, sin(x) = 0 at integer multiples
of π (0, π, 2π, 3π, etc.). The function will be at its most extreme point when sin(x) = 1 or -1. Negative displacements mean that the pendulum
is on the opposite side that it started. These equations belong to a broader type of functions that we will discuss next.

What else experiences a restoring force?

The motion of a simple pendulum is an example of Simple Harmonic Motion (SHM). By definition, SHM is when something oscillates back
and forth because it experiences a restoring force proportional to how far away it is from its equilibrium position. Spring-mass
systems, pendulums, orbits in 2 dimensions, and many other systems can be described as undergoing simple harmonic motion. The general
formula for anything that is oscillating under SHM is the equation of a wave:

x = Asin(tf)

Where x is the displacement from equilibrium
A is the amplitude or max displacement from equilibrium
t is time
f is the frequency, or the amount of cycles per unit time

You can also use a cosine version of this function by shifting the function π/2 to the right or by assuming the motion starts at its most extreme point.

For example, for a spring-mass system, the equation is identical to a pendulum and the general formula above except that the frequency is
based on the mass (m) and spring constant (k). Also, amplitude for a spring mass system is measured in distance instead of an angle for pendulums.

The restorative force for a spring mass system is the spring force as derived from Hooke’s Law (F=-kx).

Works Consulted:

Tipler, Paul A., and Gene Mosca. Physics for Scientists and Engineers. New York: W.H. Freeman, 2003

Pendulum Practice Quiz