Mass_of_a_simple_pendulum Simple_harmonic_motion

in small-angle approximation, motion of simple pendulum approximated simple harmonic motion. period of mass attached pendulum of length l gravitational acceleration



g


{\displaystyle g}

given by

t
=
2
π



l
g





{\displaystyle t=2\pi {\sqrt {\frac {l}{g}}}}

this shows period of oscillation independent of amplitude , mass of pendulum not of acceleration due gravity,



g


{\displaystyle g}

, therefore pendulum of same length on moon swing more due moon s lower gravitational field strength. because value of



g


{\displaystyle g}

varies on surface of earth, time period vary place place , vary height above sea level.

this approximation accurate small angles because of expression angular acceleration α being proportional sine of displacement angle:

−
m
g
l
sin
⁡
θ
=
i
α
,


{\displaystyle -mgl\sin \theta =i\alpha ,}

where moment of inertia. when θ small, sin θ ≈ θ , therefore expression becomes

−
m
g
l
θ
=
i
α


{\displaystyle -mgl\theta =i\alpha }

which makes angular acceleration directly proportional θ, satisfying definition of simple harmonic motion.