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 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 the Young's modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate. Find the time period of oscillation.

Moderate

A

$2 π \sqrt{\frac{m (YA + kL)}{YAk}}$
B

$\frac{1}{2 π} \sqrt{\frac{mYA + kL}{YAk}}$
C

$2 π \sqrt{\frac{YAk}{mYA + kL}}$
D

$2 π \sqrt{\frac{kL}{YAk + mYA}}$

Solution

Here, the wire can be considered as a spring of spring constant $\frac{YA}{L}$ .
$\frac{1}{k_{eq}} = \frac{1}{k_{1}} + \frac{1}{k_{2}}$
$\Rightarrow k_{eq} = \frac{k_{1} k_{2}}{k_{1} + k_{2}} = \frac{k (\frac{YA}{L})}{k + \frac{YA}{L}} = \frac{YAk}{kL + YA}$
$\Rightarrow T = 2 π \sqrt{\frac{m}{k_{eq}}} = 2 π \sqrt{\frac{m (YA + kL)}{YAk}}$

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