Answer to Question #258445 in Physics for oween

Question

A wire of length 0.50 m is fixed horizontally between two supports separated by 0.50 m. When a

mass of 8 kg hangs from the middle of the wire, the mid-point sags by 1 cm. The diameter of the

wire is 2.8 mm. Calculate the Young’s modulus of the wire.

Answer 1

Find the change in length. First, find the stretched length:

$$L_s=2\sqrt{(0.5/2)^2+0.01^2}=0.5004\text{ m}.$$

Find the tension in the wire:

$$2T\sin\arctan\frac{L_\text{sag}}{L/2}=mg,\\\space\\ T=981\text{ N}.$$

Strain:

$$\epsilon=\frac{L_s-L}{L}=\frac\sigma E,\\\space\\ E=\frac{\sigma L}{L_s-L},\\\space\\ \sigma=T/A,\\\space\\ E=\frac{TL}{A(L_s-L)},\\\space\\ E=199\text{ GPa}.$$