Question

A light rod of length $l$ pivoted at O is connected with two springs of stiffness $k_1\&k_2$ at a distance of $a$ & $l$ from the pivot respectively. A block of mass $m$ attached with the spring $k_2$ is kept on a smooth horizontal surface. Find the angular frequency of small oscillations of the block $m$.

• A. $\omega =\sqrt{\frac{k_1^2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$
• B. $\omega =\sqrt{\frac{k_2^2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$
• C. $\omega =\sqrt{\frac{k_1{k}_2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$
• D. None of these

Answer 1

The correct option is C $\omega =\sqrt{\frac{k_1{k}_2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$


Let the block be pulled towards right through a distance $x$ given as,
$x=X_B+X_{CB}....\left(i\right)$
where, $X_{CB}$ = displacement of block C relative to B

FBD of the system is shown in figure below


$X_{CB}=\frac{F}{k_2}....\left(ii\right)$
Extension in spring $1$, $X_A=\frac{F^{\prime }}{k_1}$
We know, $\frac{X_A}{X_B}=\frac{a}{l}$
[from the diagram, using similarity of triangles]
$\Rightarrow X_B=\left(\frac{F^{\prime }}{k_1}\right)\frac{l}{a}....\left(iii\right)$
$F$ & $F^{\prime }$ can be related by taking the moments of these forces about O, which yields
${\tau }_0=F^{\prime }a-Fl$
$\Rightarrow I_0\frac{d^2\theta }{d{t}^2}=F^{\prime }a-Fl$
Since the rod is light, its MOI $I_0$ about O is equal to zero.
$\Rightarrow F^{\prime }=F\left(\frac{l}{a}\right)....\left(iv\right)$
Using (iii) & (iv)
$\Rightarrow X_B=\frac{F}{k_1}{\left(\frac{l}{a}\right)}^2.....\left(v\right)$
Using (i), (ii) & (v),
$x=\frac{F}{k_1}{\left(\frac{l}{a}\right)}^2+\frac{F}{k_2}$
$\Rightarrow F=\frac{k_1{k}_2}{k_2{\left(\frac{l}{a}\right)}^2+k_1}x$
$\Rightarrow m{\omega }^{2}x=\frac{k_1{k}_2}{k_2{\left(\frac{l}{a}\right)}^2+k_1}x$
$\Rightarrow \omega =\sqrt{\frac{k_1{k}_2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$