Question

The length of a metal wire is recorded to be l₁ under a tension T₁ and l₂ under a tension T₂ . The natural length of the wire is

(1) \(\frac{l_1T^2+l_2T^2_2}{T^2_1+T^2_2}\)

(2) \(\frac{l_1T_2+l_2T_1}{T_2-T_1}\)

(3) \(\frac{l_1T_2-l_2T_1}{T_2+T_1}\)

(4) \(\frac{l_1T_2-l_2T_1}{T_2-T_1}\)

Answer 1

 (4) \(\frac{l_1T_2-l_2T_1}{T_2-T_1}\)

For a force F on a wire of length L elongated by ΔL ; the Young’s modulus Y = \(\frac{FL}{AΔL}.\) As A (cross– sectional area) and L are constants; therefore

\(\frac{F}{ΔL}=\frac{YA}{L}\) = constant or ΔL α F . Therefore (l₁ - L) and (l₂ - L) are directly proportional to T₁ and T₂ ; i.e