Question

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

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 :

$$T=2\pi \sqrt{\frac{m}{K}}=2\pi \sqrt{\frac{m}{k_1+k_2}}$$

(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,

$$a=\left(\frac{F}{m}\right)=\frac{(k_1+k_2)}{m}x$$

Time period $$\left(T\right)$$ is given by ,

$$T=2\pi \sqrt{\frac{\text{ displacement }}{\text{ acceleration }}}$$ 

$$\text{ On substituting the values of displacement and acceleration, we get: }$$ $$T=2\pi \sqrt{\frac{x}{x\frac{(k_1+k_2)}{m}}}$$ 

$$=2\pi \sqrt{\frac{m}{k_1+k_2}}$$

Required spring constant, K = k₁ + k₂

^{(c) Let K be the equivalent spring constant of the series combination.}

$$\frac{1}{K}=\frac{1}{k_1}+\frac{1}{k_2}=\frac{k_2+k_1}{k_1{k}_2}$$ 

$$\Rightarrow K=\frac{k_1{k}_2}{k_1+k_2}$$ 

$$\text{ Time period is given by, }$$ 

$$T=2\pi \sqrt{\frac{m}{K}}$$ 

$$\text{ On substituting the respective values, we get: }$$ $$T=2\pi \sqrt{\frac{m(k_1+k_2)}{k_1{k}_2}}$$