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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?

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Solution

(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=2πmK=2πmk1+k2



(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 a is given by,
a=Fm=k1+k2mx

Time period T is given by,
T=2πdisplacementaccelerationOn substituting the values of displacement and acceleration, we get:T =2πxxk1+k2m =2πmk1+k2



Required spring constant, K = k1 + k2

(c) Let K be the equivalent spring constant of the series combination.
1K=1k1+1k2=k2+k1k1k2K=k1k2k1+k2Time period is given by,T=2πmKOn substituting the respective values, we get:T=2πmk1+k2k1k2


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