# The simple pendulum

Consider a pendulum of length L with a mass m at the end displaced through an angle θ from the vertical (Figure 1). The restoring force F is the component of the weight of the bob. Therefore

F = - mg sinθ = ma giving a = - g sinθ
But for small angles sin θ tends to θc, and therefore   a = -gθ = - gx/L

where x is the distance of the bob from the midpoint of the oscillation. The acceleration is proportional to the negative of the displacement and so the pendulum therefore moves with simple harmonic motion.

The value of ω2 is g/L, and so the period of a simple pendulum is

Period of a simple pendulum (T) = 2π[L/g]1/2

(this formula is only accurate for small angles of swing, however).

An alternative treatment uses the idea of the moment of inertia of the bob (mL2) and the restoring couple (C = - mg sinθ L).

Since C = Iα we have:     - mg sin θL = Iα = mL2α

Therefore the angular acceleration (α) = - gθ/L for small θ giving the same result as the first proof.

Example problem
Calculate the length of a simple pendulum that will have a period of exactly one second. (g = 9.81 ms-2)
L = [T2g]/4π2 = [1x9.81] /39.47 = 0.25m

## The measurement of the acceleration due to gravity

A simple pendulum may be used to measure the acceleration due to gravity (g). The period is measured for a series of different values of L and a graph plotted of T2 against L.

The gradient of this graph is L/T2 and this is equal to g/4π2.

Therefore g = 4π2L/T2

From this the value of g can be found. Very accurate determinations by this method have been used in geophysical prospecting.

## Simple pendulum with large angles of swing

For large angles of swing (θ) the period of the simple pendulum is:

T = 2π[L/g(1+[1/22]sin2θ + [1.32/22.42]sin4θ+...)]1/2

although the simple formula is accurate to ±0.5 per cent for θ<15%.

A simple pendulum of length 1 m has a theoretical period of swing (using the simple formula) of 2.006 s. If the swing is now increased to 45o this becomes 2.131 s.

Student investigation
Use a pendulum at least 2 m long with a small but heavy bob to make a determination of the acceleration due to gravity. Include in your results an estimate of the accuracy, taking into account the angle of swing if you consider it to be an important factor.