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 with a time period T equal to

• A. $2\pi(m/k)^{1/2}$
• B. $2\pi\sqrt{\frac{m(YA+KL)}{YAk}}$
• C. $2\pi(mYA/kL)^{1/2}$
• D. $2\pi(mL/YA)^{1/2}$

Answer 1

Answer: (B)

$2\pi\sqrt{\frac{m(YA+KL)}{YAk}}$

Answer 2

$K_{eq}=\frac{k_1k_2}{k_1+k_2}=\frac{\frac{YA}{L}}{\frac{YA}{L}+k}=\frac{YAk}{YA+Lk}$
$T=2\pi\sqrt{\frac{m}{k_{eq}}}$
$2\pi\sqrt{\frac{m(YA+KL)}{YAk}}$
NOTE Equivalent force constant for a wire is given by $k=\frac{YA}{L}.$ Because in
case of a wire, $F=\bigg(\frac{YA}{L}\bigg)\Delta L$ and in case of spring, $F=k.\Delta x.$ Comparing these two, we find k of wire$=\frac{YA}{L}$