Simple pendulum

Simple Pendulum
Purpose
In this experiment, you will learn the physics of the simple pendulum. You will find the relationship
between the period of a pendulum and some other parameters like the length of the pendulum, the mass
of the bob or the amplitude of the oscillation. You will also verify the conservation of energy of the
pendulum.

Theory
Pendulum have been used for a long time to measure the time. A pendulum when oscillating with a small
angle, follows a harmonic motion:
𝜃(𝑡) = 𝜃0 cos (

2𝜋𝑡
)
𝑇

(4.1)

where 𝜃(𝑡) is the angle of the pendulum as a function of time with respect to the vertical, 𝜃0 is the
amplitude of the oscillation, 𝑡 is the time and 𝑇 is the period of the pendulum.

FT

Figure 1: Simple pendulum diagram.

On Figure 1, 𝐿 is the length of the pendulum, 𝐹𝑇 is the tension in the string, 𝑚𝑔 is the gravitational force
applied on the bob, 𝑥 is the horizontal displacement of the bob and ℎ is its vertaical displacement.

The period for small oscillations can be approximated by:
𝐿
𝑇 ≈ 2𝜋√
𝑔

for 𝜃0 ≪ 1 rad

(4.2)

where 𝐿 is the length of the pendulum and 𝑔 is the gravitational acceleration. As you can see, the period
is independent of the mass of the bob. For larger oscillations, you need to consider the infinite series:
𝐿
1 2
11 4
𝑇 = 2𝜋√ (1 +
𝜃0 +
𝜃 + ⋯)
𝑔
16
3072 0

(4.3)

In Eq. 4.3, 𝜃0 is in radians. In this case, the amplitude of the oscillation has an effect on the period, but it
is very small, in most cases Eq. 4.2 is accurate enough. In fact, when 𝜃0 ≈ 23𝑜 the difference is only 1%,
but it gets much bigger quickly for greater angles.
Measuring an angle is not always easy, instead we will measure the 𝑥 displacement and from it, we can
calculate the angle using:
𝑥
𝜃 = sin−1 ( )
𝐿

(4.4)

The angle should be in radians, make sure you use the right settings on your calculator.
For the last part of the experiment, you will be asked to verify the conservation of energy of your...