One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cr

Question

One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to 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 T equal to

TestHub · Accepted Answer

The wire may be treated as a spring for which force constant

$k_{1} = \frac{F o r c e}{E x t e n s i o n} = \frac{Y A}{L}$ $(∵ Y = \frac{F}{A} \times \frac{L}{Δ L})$

Spring constant of the spring $k_{2} = K$

Hence spring constant of the combination (series)

$k_{e q} = \frac{k_{1} k_{2}}{k_{1} + k_{2}} = \frac{(\frac{Y A}{L}) K}{(\frac{Y A}{L}) + K} = \frac{Y A K}{Y A + K L}$

$∵$ Time period $T = 2 π \sqrt{\frac{m}{k}} = 2 π {[\frac{(Y A + K L) m}{Y A K}]}^{\frac{1}{2}}$

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