Pendulums
Discussion
the simple pendulum
A pendulum is a mass suspended from a pivot point that is free to swing back and forth. Because the motion is oscillatory (a fancy way to say back and forth) and periodic (repeating with a characteristic time), pendulums have been used in clocks since the 17th century. Crude pendulums are cheap and easy to build — all you need is a small weight, a piece of string, and something to hang it from — and make appropriate handson devices for introductory physics courses. The arms and legs of a walking person are also pendulums, so your typical offtheshelf human comes with pendulums included.
A simple pendulum is a mathematical idealization used to approximate the behavior of real pendulums. Simple pendulums do not exist, but they're how real physics gets done. Start with the simple and build up to the real. All the mass of a simple pendulum is concentrated at a single point (called the bob) on the end of an unstretchable, incompressible, massless rod connected to a frictionless pivot that does not move. The bob of a simple pendulum traces back and forth over the arc of a circle with the pivot at the center. A stationary pendulum of any sort will rest in an equilibrium position with its center of mass below the pivot.
Two forces act on the bob a simple pendulum whether it is moving or not — weight (the force of gravity) and tension. (Air resistance is ignored, as is often the case. Poor air resistance. Always being ignored.) In the equilibrium position, when the pendulum is at rest, the two forces are equal and opposite. Not very interesting.
Since the whole pendulum is held fixed at the pivot (the noun), any attempt to move the pendulum side to side causes the whole thing to pivot (the verb) with an angular displacement θ. Now things get interesting.
This is a problem where a radialtangential coordinate system works better than the typical verticalhorizontal one. Tension points radially inward toward the pivot and is considered the "good" vector. Weight is the "bad" vector and must be broken up into components — one radial (mg sin θ) and one tangential (mg cos θ).
The radial component and the tension contribute to the centripetal force. Tension is the positive one since it points toward the center. The radius (r) in the centripetal acceleration equation could be replaced with the length of the pendulum (ℓ) since they are the same thing. The tangential velocity (v) can also be written as a derivative for those who like calculus. There's no real reason to do that now other than because it looks cool. Actually, there's no reason to write any of this now other than for a sense of completeness.
∑F_{r} =  ma_{c}  
T − mg cos θ = 


T − mg cos θ = 


The tangential component is the more interesting one. It's described a restoring force because it increases with increasing displacement and acts to push the bob back toward its equilibrium position. For that reason, it gets a negative sign. When displacement is positive, it points in the negative direction. When displacement is negative, it points in the positive direction. In informal language, it does the opposite. Now the tangential acceleration really should be written in fancy calculus notation. It's more useful that way. Also the mass cancels out, which is interesting.
∑F_{t} =  ma_{t}  
− mg sin θ = 


− g sin θ = 


This is a second order differential equation that is extremely difficult to solve. I have never done it myself. You might think that's the end of this discussion, but you've forgotton one thing. This is a simple pendulum. It's already full of assumptions that aren't true (point masses, unstretchable strings, etc.). One more simplification couldn't hurt.
small angle approximation
Go back to the definition of sine as the ratio of the side opposite (x) the angle (θ) to the hypotenuse (r). Here's a (possibly new) fact for you known as the small angle approximation: the opposite side length (x) approaches the arc length (s) as the angle (θ) approaches zero (0).
θ → 0  ⇒  x → s 
This means that…
sin θ =  x 
r 
can be replaced with…
sin θ ≈  s 
ℓ 
…for a simple pendulum when the angle is small. That gives us a differential equation that is much easier to solve.
− g sin θ = 







We need a function whose second derivative is itself with a minus sign. We have two options: sine and cosine. Either one is fine since they're basically identical functions with a 90° phase shift between them. Without loss of generality, I'll choose sine with an arbitrary phase angle (φ) that could equal 90° if we let it. Or it equal 0° or some other angle. The other parameters in a generic sine function are displacement amplitude (s_{0}) and angular frequency (ω).
The basic method I've started is called "guess and check". My guess is that the function looks like a generic sine function…
s = s_{0} sin(ωt + φ)
and the check is to pop it back into the differential equation and see what happens.
d^{2}  s_{0} sin(ωt + φ)  = −  g  s_{0} sin(ωt + φ) 
dt^{2}  ℓ  
− ω^{2}s_{0} sin(ωt + φ)  = −  g  s_{0} sin(ωt + φ)  
ℓ  
ω^{2}  =  g  
ℓ 
Basically everything cancels but one of my three parameters — angular frequency.
ω = √  g 
ℓ 
I don't think in angular frequencies. They're too abstract. I want the frequency — frequency with no adjective in front. Something that uses hertz [Hz] as the unit.
f =  ω  =  1  √  g 
2π  2π  ℓ 
Actually, I don't even want that. Pendulums tend to be slow beasts. I'd prefer the reciprocal of frequency — period. The period of a simple pendulum with small angle amplitudes is given by the following equation…
T = 2π√  ℓ 
g 
Where…
T =  period [s], the time to complete one cycle of motion 
π =  a mathematical constant [unitless] 
ℓ =  length [m], measured from the suspension point to the mass 
g =  gravity [m/s^{2}], the gravitational field strength in the place where the pendulum is doing its thing 
What this says about simple pendulums…
 The period of a simple pendulum is independent of it mass. Light bob, heavy bob, no difference. This should make sense since the acceleration of a freely falling body is independent of its mass. For practical reasons, however, mass may be worth considering when air resistance is added in. Heavy masses are less affected by aerodynamic drag than lighter ones since acceleration is inversely proportional to mass. Do you want your pendulum to keep plowing through the air? Then make it massive.
 The period of a simple pendulum is proportional to the square root of its length (T ∝ √ℓ). Longer pendulums take longer to swing back and forth once than short ones. Doubling the period of a simple pendulum requires a quadrupling of its length. This should make intuitive sense on some level. Long legs take longer to swing back and forth than short legs do. A elephant takes fewer footsteps than a mouse would at the same speed.
 The period of a simple pendulum is inversely proportional to the square root of the local gravitational field strength (T ∝ 1/√g). Doesn't quite roll off the tongue nicely, does it? Quadrupling gravity would halve the period of a simple pendulum. Not really a situation I could see myself encountering. Local gravity doesn't vary much across the surface of the Earth — at least as far as direct human perception is concerned — but it matters when pendulums are how you keep accurate track of time.
What this doesn't say about simple pendulums…
 The period of a simple pendulum increases with increasing amplitude in a way that is difficult to describe, difficult to explain, and frequently unimportant.
large angle correction
The equation for the period of a simple pendulum that accounts for large angles is the small angle approximation…
T = 2π√  ℓ 
g 
multiplied by an infinite sum of ever smaller corrections. The first 5 terms are shown below. I will not attempt to derive this equation.
T = 2π√  ℓ  ⎡ ⎢ ⎣ 
1 +  ⎛ ⎜ ⎝ 
1  ⎞^{2} ⎟ ⎠ 
sin^{2}  ⎛ ⎜ ⎝ 
θ  ⎞ ⎟ ⎠ 
g  2  2 
+  ⎛ ⎜ ⎝ 
1 · 3  ⎞^{2} ⎟ ⎠ 
sin^{4}  ⎛ ⎜ ⎝ 
θ  ⎞ ⎟ ⎠ 
2 · 4  2 
+  ⎛ ⎜ ⎝ 
1 · 3 · 5  ⎞^{2} ⎟ ⎠ 
sin^{6}  ⎛ ⎜ ⎝ 
θ  ⎞ ⎟ ⎠ 
2 · 4 · 6  2 
+  ⎛ ⎜ ⎝ 
1 · 3 · 5 · 7  ⎞^{2} ⎟ ⎠ 
sin^{8}  ⎛ ⎜ ⎝ 
θ  ⎞ ⎟ ⎠ 
+ …  ⎤ ⎥ ⎦ 
2 · 4 · 6 · 8  2 
Since every term in this infinite beast is positive, the true period of a simple pendulum will always be greater than that calculated using the small angle approximation. The correction adds <0.1% up to 7°, <1% up to 22°, <10% up to 69°, and about 18% at 90°.
If even larger angles are allowed, then the large angle correction maxes out at +116% for pendulums with an angle of release of 180°.
Another version of the infinite sum of corrections is shown below. This one starts with the equation derived above and replaces all the sine functions with their Taylor expansions (each one an infinite sum unto itself). Collect like terms, remove common factors, and this is what you get…
T = 2π√  ℓ  ⎡ ⎢ ⎣ 
1 +  1  θ^{2} 
g  16 
+  11  θ^{4} 
3,072 
+  173  θ^{6} 
737,280 
+  22,931  θ^{8}  + …  ⎤ ⎥ ⎦ 
1,321,205,760 