Examples Simple_harmonic_motion

1 examples

1.1 mass on spring
1.2 uniform circular motion
1.3 mass of simple pendulum
1.4 scotch yoke

examples

the following physical systems examples of simple harmonic oscillator.

mass on spring

a mass m attached spring of spring constant k exhibits simple harmonic motion in closed space. equation describing period

t
=
2
π



m
k





{\displaystyle t=2\pi {\sqrt {\frac {m}{k}}}}

shows period of oscillation independent of both amplitude , gravitational acceleration. above equation valid in case when additional constant force being applied on mass, i.e. additional constant force cannot change period of oscillation.

uniform circular motion

simple harmonic motion can considered one-dimensional projection of uniform circular motion. if object moves angular speed ω around circle of radius r centered @ origin of xy-plane, motion along each coordinate simple harmonic motion amplitude r , angular frequency ω.

mass of simple pendulum

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.

scotch yoke

a scotch yoke mechanism can used convert between rotational motion , linear reciprocating motion. linear motion can take various forms depending on shape of slot, basic yoke constant rotation speed produces linear motion simple harmonic in form.