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Question

Four massless springs whose force constants are 2k,2k,k and 2k, respectively, are attached to a mass M kept on a frictionless horizontal plane (as shown in figure). If the other ends of the springs are fixed and the mass M is displaced , then the frequency of oscillation of the system is


A
12πk4M
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B
12π4kM
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C
12πk7M
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D
12π7kM
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Solution

The correct option is B 12π4kM
The two springs on left side to the mass M are having spring constant of 2k each. From the figure we can say that they are in series combination;
Equivalent spring constant is keq=k1k2k1+k2=2k (2k)2k+2k=k ..........(1).
The two springs on right hand side of mass M are in parallel combination. Their effective spring constant is k′′eq=(k+2k)=3k ......(2).
From (1) and (2)
Equivalent spring constants of value k and 3k are in parallel combination.
Thus, the effective spring constant of the system of springs is given by keq=keq+k′′eq=k+3k=4k
Angular frequency of the oscillation of mass M is given by ω=keqM=4kM
But ,
ω=2πf=4kMf=12π4kM
Thus, option (b) is the correct answer.

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