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

Q: 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

b (b) The wire may be treated as a string for which force constant ${k_1} = \frac{{Force}}{{Extension}} = \frac{{YA}}{L}\;\;\;\left( {\because Y = \frac{F}{A} \times \frac{L}{{\Delta L}}} \right)$ 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 $T = 2\pi \sqrt {\frac{m}{k}} = 2\pi {\left[ {\frac{{(YA + KL)m}}{{YAK}}} \right]^{1/2}}$

SECTION - A [PHYSICS - MCQ]

GSEB - English Medium

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

A$2\pi \left( {\frac{m}{K}} \right)$
B$2\pi {\left\{ {\frac{{(YA + KL)m}}{{YAK}}} \right\}^{1/2}}$
C$2\pi \frac{{mYA}}{{KL}}$
D$2\pi \frac{{mL}}{{YA}}$

IIT 1993, Diffcult

STD 11 Science
NEET
STD 11 - 13. oscillations
Physics

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