The differential equation of all âSimple Harmonic Motionsâ of given period (2π/n) is

Question

The differential equation of all âSimple Harmonic Motionsâ of given period (2π/n) is (A) (d2x/dt2)+nx=0 (B) (d2x/dt2)+n2x=0 (C) (d2x/dt2)-n2x=0 (D) (d2x/dt2)+(1/n2)x=0

Tardigrade Admin · Accepted Answer

The displacement x for all S.H.M. is given by x=a cos(nt+b) ⇒ (dx/dt)=-na sin(nt+b) ⇒ (d2x/dt2)=-n2a cos(nt+b) ⇒ (d2x/dt2)=-n2x ⇒ (d2x/dt2)+n2x=0

Q. The differential equation of all ‘Simple Harmonic Motions’ of given period $\frac{2 π}{n}$ is

$\frac{d ^{2} x}{d t ^{2}} + n x = 0$

$\frac{d ^{2} x}{d t ^{2}} + n^{2} x = 0$

$\frac{d ^{2} x}{d t ^{2}} - n^{2} x = 0$

$\frac{d ^{2} x}{d t ^{2}} + \frac{1}{n ^{2}} x = 0$

The displacement $x$ for all $S . H . M .$ is given by
$x = a cos (n t + b)$
$\Rightarrow \frac{d x}{d t} = - na s in (n t + b)$
$\Rightarrow \frac{d ^{2} x}{d t ^{2}} = - n^{2} a cos (n t + b)$
$\Rightarrow \frac{d ^{2} x}{d t ^{2}} = - n^{2} x$
$\Rightarrow \frac{d ^{2} x}{d t ^{2}} + n^{2} x = 0$

Q. The differential equation of all ‘Simple Harmonic Motions’ of given period n2π​ is

dt2d2x​+nx=0

dt2d2x​+n2x=0

dt2d2x​−n2x=0

dt2d2x​+n21​x=0

The displacement x for all S.H.M. is given by x=acos(nt+b) ⇒dtdx​=−nasin(nt+b) ⇒dt2d2x​=−n2acos(nt+b) ⇒dt2d2x​=−n2x ⇒dt2d2x​+n2x=0

Q. The differential equation of all ‘Simple Harmonic Motions’ of given period $\frac{2 π}{n}$ is

$\frac{d ^{2} x}{d t ^{2}} + n x = 0$

$\frac{d ^{2} x}{d t ^{2}} + n^{2} x = 0$

$\frac{d ^{2} x}{d t ^{2}} - n^{2} x = 0$

$\frac{d ^{2} x}{d t ^{2}} + \frac{1}{n ^{2}} x = 0$