## Maximum Force Experienced By a Body Experiencing Simple Harmonic Motion

Simple harmonic motion is very common, In fact, almost all vibrations, if small enough, are approximately simple harmonic. We are often especially interested in the maximum force to which a body may be subject during simple harmonic motion. A safety margin is built into the system to ensure that the system will operate safely up to a required lifetime.The maximum acceleration experienced by a body undergoing simple harmonic motion is

\[a_{MAX} = \omega^2 A\]

where \[A , \: \omega\]

are the amplitude and the angular frequency of the motion.Suppose a body of mass 0.3 kg executes simple harmonic motion with amplitude

\[A=2.1 m\]

and frequency \[f=20 Hz\]

.Then

\[\omega = 2 \pi f = 2 \pi \times 20 =40 \pi\]

.Then

\[a_{MAX} = \omega^2 A = (40 \pi)^2 \times 2.1=33161 m/s^2\]

Now use Newtons Second Law

\[F=ma\]

.The maximum force is

\[F_{MAX} =0.3 \times 33162=9948.6 N\]

.