Question
Obtain the differential equation of linear simple harmonic motion.
✓
Answer
Differential equation of linear $\text{S.H.M}$:
Let a particle of mass ‘m’ undergo $\text{S.H.M}$ about its mean position $ O$.
At any instant $'t\ ',$ displacement of the particle be $'x\ '$ as shown in the following figure.
By definition $, F = - kx ....(1)$
where $k$ is force constant
The acceleration of the particle is given by,
$a=\frac{d v}{d t}=\frac{d\left(\frac{d x}{d t}\right)}{d t}=\frac{d^2 x}{d t^2}$
According to Newton’s second law of motion,
$F = ma$
$\therefore F=m\left(\frac{d^2 x}{d t^2}\right) .....(2)$
From equations $(1)$ and $(2),$
$m\left(\frac{d^2 x}{d t^2}\right)=-k x$
$\therefore \frac{d^2 x}{d t^2}=-\frac{k}{m} x$
$\therefore \frac{d^2 x}{d t^2}+\frac{k}{m} x=0 ....(3)$
where, $\frac{k}{m}=\omega^2=$ cons $\tan t$
$\therefore \frac{d^2 x}{d t^2}+\omega^2 x=0 ....(4)$
Equations $(3)$ and $(4)$ represent differential equations of linear $\text{S.H.M.}$
Let a particle of mass ‘m’ undergo $\text{S.H.M}$ about its mean position $ O$.
At any instant $'t\ ',$ displacement of the particle be $'x\ '$ as shown in the following figure.
By definition $, F = - kx ....(1)$
where $k$ is force constant
The acceleration of the particle is given by,
$a=\frac{d v}{d t}=\frac{d\left(\frac{d x}{d t}\right)}{d t}=\frac{d^2 x}{d t^2}$
According to Newton’s second law of motion,
$F = ma$
$\therefore F=m\left(\frac{d^2 x}{d t^2}\right) .....(2)$
From equations $(1)$ and $(2),$
$m\left(\frac{d^2 x}{d t^2}\right)=-k x$
$\therefore \frac{d^2 x}{d t^2}=-\frac{k}{m} x$
$\therefore \frac{d^2 x}{d t^2}+\frac{k}{m} x=0 ....(3)$
where, $\frac{k}{m}=\omega^2=$ cons $\tan t$
$\therefore \frac{d^2 x}{d t^2}+\omega^2 x=0 ....(4)$
Equations $(3)$ and $(4)$ represent differential equations of linear $\text{S.H.M.}$
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