Simple pendulums are sometimes used as an example of simple harmonic motion, SHM, since their motion is periodic. They also fit the criteria that the bob's velocity is maximum as it passes through equilibrium and its acceleration is minimal while at each endpoint. But a deeper understanding of the behavior of the bob will show us that pendulums do not truly fit the SHM model.

 

To begin our analysis, we will start with a study of the properties of force and acceleration in a simple pendulum by examining a freebody diagram of a pendulum bob.

 

 

As the pendulum swings, it is accelerating both centripetally, towards the point of suspension and tangentially, towards its equilibrium position. It is its linear, tangential acceleration that connects a pendulum with simple harmonic motion.

 

 

 

The weight component, mg sinθ, is accelerating the mass towards equilibrium along the arc of the circle. This component is called the restoring force of the pendulum.

 

F_restoring = ma_tangential
mg sinθ = ma_tangential

 

To strictly qualify as SHM this restoring force should be directly proportional to the bob’s linear displacement from equilibrium along the length of the chord.

 

 

 

Geometrically, the arc length, s, is directly proportional to the magnitude of the central angle, θ, according to the formula s = rθ. In our diagram the radius of the circle, r, is equal to L, the length of the pendulum. Thus, s = Lθ, where θ must be measured in radians. Substituting into the equation for SHM, we get

 

F_restoring= - ks
mg sinθ = - k(Lθ)

 

Solving for the "spring constant" or k for a pendulum yields 

 

mg sinθ = k(Lθ)

k = mg sinθ / Lθ

 

When an angle is expressed in radians, mathematicians generally represent the angle with the variable x instead of θ. Note that the value of sin x approximates the value of x for small angles; that is,

 

 

Or, equivalently, for x equal to small values you can see from this power series that the value of sin x would approach that of x.

 

 

Using this relationship allows us to reduce our expression for the pendulum's "spring constant" to

 

k = mg / L

 

Substituting this value for k into the SHM equation for the period of an oscillating system results in

 

 

Before leaving this lesson, let's examine our formula's error for a pendulum released at an angle of 10º.

 

10º = 0.1745 radians

sin(0.1745) = 0.1736

 

 

 

Our calculated value for the period will be 0.26% too high. An error that most would readily accept in lieu of using a more complicated formula.

 

Take a few moments and use this physlet to investigate how the period of a pendulum is impacted by its length and its initial displacement.