Q01)
Consider the motion of a simple pendulum shown in the Figure 01 and answer the following
questions given in below.
(a) Describe a suitable assumption set that you considered investigating the lost less motion
of the simple pendulum.
โข Negligible air resistance: Assume that the pendulum moves in a vacuum or an
environment with minimal air resistance.
โข Perfectly elastic support: Assume the support (pivot point) of the pendulum is
frictionless and perfectly elastic, with no energy losses.
โข Small angle approximation: Assume the angle of displacement is small, allowing the
use of the small-angle approximation sin(ฮธ) โ ฮธ.
(b) Derive a differential equation to describe the lost-less motion of the simple pendulum.
๐ผ๐2๐
๐2๐ = โ๐๐๐ฟsin๐
where:
I is the moment of inertia of the pendulum bob,
ฮธ is the angular displacement,
t is time,
m is the mass of the bob,
g is the acceleration due to gravity,
L is the length of the pendulum.
(c) Obtained the solution of the differential equation.
The above differential equation is a nonlinear second-order differential equation.
For small-angle approximation (sinฮธโฮธ), it can be linearized and solved. The solution is
typically in the form:
๐(๐ก) = ๐0cos(โ๐
๐ฟ๐ก + ๐)
