Question

A 4.00 m-long steel cable is used to support a church chandelier. After the 226-kg chandelier is hung from the ceiling, the steel cable is lengthened by 3.00 mm. Calculate the diameter of the cable in millimeters. (The Young’s modulus for steel is ????????. ???????????????? × ???????????????????????????????? ????????/????????????????.)

Answer 1

Answer:

4.22 mm

Explanation:

E = Young’s modulus for steel = 210 GPa (generally)

$\Delta L$ = Change in length = 3 mm

$L_0$ = Original length = 4 m

A = Area of cable

g = Acceleration due to gravity = 9.81 m/s²

r = Radius of cable

d = Diameter = 2r

m = Mass of chandelier = 226 kg

$\epsilon$ = Longitudinal strain = $\frac{\Delta L}{L_0}$

Uniaxial stress is given by

$\sigma=E\epsilon\\\Rightarrow \sigma=210\times 10^9 \frac{3\times 10^{-3}}{4}\\\Rightarrow \sigma=157500000\ Pa$

$\sigma=\frac{F}{A}\\\Rightarrow \sigma=\frac{mg}{\pi r^2}\\\Rightarrow 157500000=\frac{226\times 9.81}{\pi r^2}\\\Rightarrow r=\sqrt{\frac{226\times 9.81}{157500000\times \pi}}\\\Rightarrow r=0.00211\ m\\\Rightarrow d=2r\\\Rightarrow d=2\times 2.11=4.22\ mm$

The diameter of the cable is 4.22 mm