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,

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

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

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

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

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

Required spring constant, K = k₁ + k₂

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

\[\frac{1}{K} = \frac{1}{k_1} + \frac{1}{k_2} = \frac{k_2 + k_1}{k_1 k_2}\] 

\[ \Rightarrow K = \frac{k_1 k_2}{k_1 + k_2}\] 

\[\text { Time  period  is  given  by, } \] 

\[T = 2\pi\sqrt{\frac{m}{K}}\] 

\[\text { On  substituting  the  respective  values,   we  get: } \] \[T = 2\pi\sqrt{\frac{m\left( k_1 + k_2 \right)}{k_1 k_2}}\]