Simple Harmonic Motion

Hooke’s Law

When certain elastic objects are stretched or compressed by a displacement ss, the restoring force is directly proportional to the displacement.

F=−ksF=-ks

FF is the restoring force, ss is the displacement,kk is a constant

Simple Harmonic Motion

If a body:

  • Has its acceleration directly proportional to its distance from a fixed point on its path.

  • Has its acceleration always directed towards that point.

a=−w2sa=-w^2s

aa is the acceleration, ss is the displacement, ww is a constant

If a system obeys Hooke’s Law:

  • F=−ksF=-ks

  • ma=−ksma=-ks (F=maF=ma )

  • a=−w2sa=-w^2s (ω2=km\omega^2=\frac{k}{m} )

The system executes simple harmonic motion.

Examples of Simple Harmonic Motion

  • Prong on a vibrating tuning fork

  • Tides coming in and out

  • Pendulum with small angle of a swing

Period of motion:

T=2πωT=\frac{2\pi}{\omega}

Frequency and Period

T=1fT=\frac{1}{f} or f=1Tf=\frac{1}{T}

TT is period, ff is frequency

Simple Pendulum

T=2πlgT=2\pi\sqrt{\frac{l}{g}}

TT is period of the pendulum, ll is length of the pendulum, gg is acceleration due to gravity

  • g=4π2lT2g=4\pi^2\frac{l}{T^2}

  • l⠀α⠀T2l⠀\alpha⠀T^2