Question
From the definition of linear SHM, derive an expression for the angular frequency of a body performing linear SHM.
✓
Answer
When a body of mass $m$ performs linear SHM, the restoring force on it is always directed towards the mean position and its magnitude is directly proportional to the magnitude of the displacement of the body from the mean position. Thus, if $\vec{F}$ is the force acting on the body when its displacement from the mean position is $\vec{x}$,
$
\vec{F}= m =- k \vec{x}
$
where the constant $k$, the force per unit displacement, is the force constant.
Let $\frac{k}{m}=\omega^2$, a constant. $m$
$\therefore$ Acceleration, $a =-\frac{k}{m} x =-\omega^2 x$
$\therefore$ The angular frequency
$
\omega=\sqrt{\frac{k}{m}}=\sqrt{\left|\frac{a}{x}\right|}
$
$
=\sqrt{\text { acceleration per unit displacement }}
$
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