First slide

Simple hormonic motion

Question

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

$2 π \sqrt{(m / k)}$
B

$2 π {[\frac{(YA + kL) m}{(YAk)}]}^{1 / 2}$
C

$2 π \sqrt{(mYA / kL)}$
D

$2 π \sqrt{(m L / Y A)}$

Solution

Let us beat the wire as a strin8. The force constant is given by
$k_{1} = \frac{force}{extension} = \frac{YA}{L} (∵ Y = \frac{F}{A} \times \frac{L}{ΔL} or \frac{F}{ΔL} = \frac{YA}{L}) The wire of force constant k_{1} and spring of force constant k are in series . The combi n ed force constan t k_{2}, is given by k_{2} = (\frac{k_{1} k}{k_{1} + k}) = \frac{(YA / L) (k)}{(YA / L) + k} = (\frac{YAk}{YA + kL}) now T = 2 π \sqrt{(\frac{m}{k_{2}})} = 2 π {[\frac{(YA + kL) m}{YAk}]}^{1 / 2} \frac{_{}}{}$

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