Question

Find the time period of the oscillation of mass m in figures 12−E4 a, b, c. What is the equivalent spring constant of the pair of springs in each case?

Figure

Answer 1

(a) Spring constant of a parallel combination of springs is given as,
K = k₁ + k₂ (parallel)
Using the relation of time period for S.H.M. for the given spring-mass system, we have:
$T=2\mathrm{\pi }\sqrt{\frac{m}{K}}=2\mathrm{\pi }\sqrt{\frac{m}{k_1+k_2}}$



(b) Let x be the displacement of the block of mass m, towards left.
Resultant force is calculated as,
F = F₁ + F₂ = (k₁ + k₂)x

Acceleration $\left(a\right)$ is given by,
$a=\left(\frac{\mathrm{F}}{m}\right)=\frac{\left(k_1+k_2\right)}{m}x$

Time period $\left(T\right)$ is given by,
$$\begin{gathered} \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{displacement}}{\mathrm{acceleration}}} \\ \mathrm{On}\mathrm{substituting}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{displacement}\mathrm{and}\mathrm{acceleration},\mathrm{we}\mathrm{get}: \\ T=2\mathrm{\pi }\sqrt{\frac{x}{x{\displaystyle \frac{\left(k_1+k_2\right)}{m}}}} \\ =2\mathrm{\pi }\sqrt{\frac{m}{k_1+k_2}} \end{gathered}$$



Required spring constant, K = k₁ + k₂

(c) Let K be the equivalent spring constant of the series combination.
$$\begin{gathered} \frac{1}{K}=\frac{1}{k_1}+\frac{1}{k_2}=\frac{k_2+k_1}{k_1{k}_2} \\ \Rightarrow K=\frac{k_1{k}_2}{k_1+k_2} \\ \mathrm{Time}\mathrm{period}\mathrm{is}\mathrm{given}\mathrm{by}, \\ T=2\mathrm{\pi }\sqrt{\frac{m}{K}} \\ \mathrm{On}\mathrm{substituting}\mathrm{the}\mathrm{respective}\mathrm{values},\mathrm{we}\mathrm{get}: \\ T=2\mathrm{\pi }\sqrt{\frac{m\left(k_1+k_2\right)}{k_1{k}_2}} \end{gathered}$$