Question

Rod is massless and of length $\ell$. Find angular frequency $\omega$ (in $s^{-1}$ ) of small oscillation of the block of mass $m$, the hinge is frictionless. $k_1,k_2,k_3$ are spring constants of the shown springs. Ignore gravity. ( $k_1=k_2=10N/m,k_3=\frac{15}{4}N/m,m=250gm)$

Answer 1

Answer: 8.00

$F=-k_{1}x-k_2\left(x-\frac{\ell }{2}\theta \right)$
$k_2\left(x-\frac{\ell }{2}\theta \right)\frac{\ell }{2}=k_3\ell \theta \ell$
$k_2\left(x-\frac{\ell }{2}\theta \right)=2k_3\ell \theta$
$\left(\frac{k_2}{2}+2k_3\right)\ell \theta =k_{2}x$
$F=-k_{1}x-k_{2}x+\frac{k_2}{2}\ell \theta$
$F=-k_{1}x-k_{2}x+\frac{k_2}{2}\left(\frac{k_{2}x}{\frac{k_2}{2}+2k_3}\right)$
$F=-10x-10x+5\left(\frac{10x}{5+\frac{15}{2}}\right)$
$F=-20x+4x$
$F=-16x$
$\Rightarrow a=-\frac{16}{m}x$
$\omega =\sqrt{\frac{16}{m}}$
$\omega =\sqrt{\frac{16}{\frac{1}{4}}}$
$\omega =8rad/s$