Question

A light pulley is suspended at the lower end of a spring of constant $k_1$, as shown in figure. An inextensible string passes over the pulley. At one end of string a mass m is suspended, the other end of the string is attached to another spring of constant $k_2$. The other ends of both the springs are attached to rigid supports, as shown. Neglecting masses of springs and any friction, find the time period of small oscillations of mass m about equilibrium position?

Answer 1

Let F be the extra tension in the string, when the block is displaced x from its mean position.
Extension in spring-2 is
$x_2=\frac{F}{k_2}$
Extension is spring-l is
$x_1=\frac{2F}{k_1}$
$x=2x_1+x_2=\frac{4F}{k_1}+\frac{F}{k_2}$
Extra tension F will become restoring force for the block. Therefore above equation can be written as,
$F=-\left(\frac{1}{\frac{4}{k_1}+\frac{1}{k_2}}\right)x=\left(\frac{k_1{k}_2}{4k_2+k_1}\right)x$
or $k_e=\frac{k_1{k}_2}{4k_2+k_1}$
$T=2\pi \sqrt{\frac{m}{k_e}}$
$=2\pi \sqrt{\frac{m(4k_2+k_1)}{k_1{k}_2}}$