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Question

A body of mass m hangs from a smooth fixed pulley P1 by an inextensible string fitted with the springs of stiffness constants k1 and k3 The string passes over a smooth light pulley P2 ,  which is connected with another ideal spring of stiffness constant k2. Find the period of small oscillations of the body.



Your Answer
A
T=2π(m(2k1+4k2+2k3))12
Correct Answer
B
T=2π(m(1k1+4k2+1k3))12
Your Answer
C
T=2π(m(2k1+2k2+2k3))12
Your Answer
D
T=2π(m(1k1+1k2+1k3))12

Solution

The correct option is D T=2π(m(1k1+4k2+1k3))12
Let x1,x2  and x3 be the elongations of the springs k1,k2 and k3 respectively. 


Using constraint relations we find the total displacement of m as 
x=x1+2x2+x3    ........(1)

Spring(3) experiences tension force due to mass m
From this, we get ,
k3x3=mgx3=mgk3
Since, a string is connected between the two springs, the tension force remains same .  So, spring (1) experiences same tension force due to mass m
x1=mgk1
Tension experienced by spring (2) is given by 2mg=k2x2x2=2mgk2
Using these values in equation (1) we obtain
x=mg[1k1+4k2+1k3]    .......(2)
Time period of oscillation  T=2πdisplacementacceleration=2πxg
Using (2) in the above equation we get,
T=2π(m(1k1+4k2+1k3))12
Hence, option (b) is the correct answer. 

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