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Question

One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to a massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross – section and Young’s modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period given by

A
2πmk
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B
2π[(YA+KL)mYAK]12
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C
2π(mYAKL)
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D
2π(mLYA)
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Solution

The correct option is B 2π[(YA+KL)mYAK]12
If the wire extends by an amount x when a force F is applied to it, then
y=FAxL=FLAx or F = (YAL)X = kx; k2 = YAL
Thus the force constant of the wire is k. if K is the forece constant of the spring, then the force constant of the series combination of the wire and the spring is given by
1k=1k1+1k2
Ork=k1k2(k1+k2)=.........(i)
The time period of the combination is
T=2πmk
Using (i) in (ii) we find that the correct choice is (b)

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