Simple Pendulum f

1. The Simple Pendulum
Simple Pendulum: there is a point mass that is attached to a string of negligible mass
that does not stretch. The string is attached to a support system that is frictionless.
The motion of the simple pendulum is periodic, as the pendulum swings back and
forth in one direction due to the force of gravity. This is only when it is released
from a certain vertical height.
Above is a depiction of a simple pendulum where the green ball is the mass that is
attached to the string of length L.
The displacement (s) is represented and related to the angular displacement as
follows:
𝑠 = 𝐿𝜃
Note:
 Displacement to the right of the equilibrium position is positive
 Displacement to the left of the equilibrium position is negative
Forces:
 Acting on the mass  mg
 Tension from the string T
Axis Systems:
 Radial axis along the length of the string
 Tangential axis tangent to the circular motion of the mass
 Both axis are perpendicular to each other and their directions change as the
mass oscillates

2. The Radial Axis:
 The mass (green ball) makes an angle 𝜃 with the radial axis
 The mass along the radial axis is mgcos 𝜃 and it points away from the
suspension point
𝑇 − 𝑚𝑔𝑐𝑜𝑠𝜃 = 0
𝑇 = 𝑚𝑔𝑐𝑜𝑠𝜃
The Tangential Axis
 The weight is mgsin𝜃
 Using Newton’s 2nd law of motion for the tangential component
𝐹𝑛𝑒𝑡, 𝑡 = −𝑚𝑔𝑠𝑖𝑛𝜃
= 𝑚𝑎
 Then where a denotes the acceleration and 𝜃 =
𝑠
𝐿
𝑎 = −𝑔𝑠𝑖𝑛𝜃
= −𝑔𝑠𝑖𝑛 (
𝑠
𝐿
)
Note:
 Acceleration is proportional to sin(s/L) not to displacement (s)
Function for small-angle approximation is as follows
𝑠𝑖𝑛𝑥 = 𝑥
Thus:
sin (
𝑠
𝐿
) =
𝑠
𝐿
Leading to:
𝑎 = −(
𝑔
𝐿
) 𝑠
The mass is proportional to the displacement and opposite in sign thus the angular
frequency is obtained by comparing the above equation with the standard equation
for simple harmonic motion: 𝜔 = √
𝑔
𝐿
The period and frequency of the oscillation is then given by
𝑇 =
2𝜋
𝜔
= 2𝜋√
𝐿
𝑔
and
𝑓 =
𝜔
2𝜋
=
1
2𝜋
√
𝑔
𝐿
Period:
 Period of a simple pendulum depends on the length of the pendulum

3.  Acceleration is due to gravity
Question to check you knowledge:
1. What is the tangential acceleration of the mass (green ball) when 𝜃 = 𝜋?
a. –g
b. –𝜋
c. 2𝜋
d. 0
Answer:
D: 0
Equation used: 𝑎 = −𝑔𝑠𝑖𝑛𝜃
sin( 𝜃) = 0
𝑎 = −𝑔(0)
𝑎 = 0