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 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 T equal to

Moderate

A

$2 π \sqrt{\frac{m}{k}}$
B

$2 π \sqrt{\frac{m (Y A + k L)}{Y A k}}$
C

$2 π \sqrt{\frac{m Y A}{k L}}$
D

$2 π \sqrt{\frac{m L}{Y A}}$

Solution

Wire is equivalent to a spring of spring constant, $k^{'} = \frac{A Y}{L}$
$k^{'} and k are in series,$
$k_{eq} = \frac{k^{'} k}{k^{'} + k} = \frac{(A Y / L) k}{(\frac{A Y}{L}) + k} = \frac{A Y k}{A Y + k L}$
$T = 2 π \sqrt{\frac{m}{k_{eq}}} = 2 π \sqrt{\frac{m (A Y + k L)}{A Y k}}$

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