## Question

One end 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

### Solution

If the wire extends by an amount *x *when a force *F* is applied to it, then

Thus the force constant of the wire is *k*. If *K* is the force constant of the spring, then the force constant of the series combination of the wire and the spring is given by

the time period of the combination is

Using (i) in (ii) we find that

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