First slide

Applications of SHM

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

Moderate

A

$2 π (\frac{m}{K})$
B

$2 π {\{\frac{(YA + KL) m}{YAK}\}}^{1 / 2}$
C

$2 π \frac{mYA}{KL}$
D

$2 π \frac{mL}{YA}$

Solution

The wire may be treated as a string for which force constant $k_{1} = \frac{Force}{Extension} = \frac{YA}{L} (∵ Y = \frac{F}{A} \times \frac{L}{ΔL})$
Spring constant of the spring $k_{2} = K$
Hence spring constant of the combination (series)
$k_{eq} = \frac{k_{1} k_{2}}{k_{1} + k_{2}} = \frac{(YA / L) K}{(YA / L) + K} = \frac{YAK}{YA + KL}$
$∵$ Time period $T = 2 π \sqrt{\frac{m}{k}} = 2 π {[\frac{(YA + KL) m}{YAK}]}^{1 / 2}$

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