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