Question

an lvdt is used to measure the axial deformation of the simple bar of fig. 10.3. the uncertainty in this measurement is ±0.025 mm. a steel bar is to be used to measure a force of 1100 n with an uncertainty of 1 percent. determine a suitable diameter and length of rod for this measurement, assum- ing the rod dimensions will be known exactly

Answer 1

$F=\frac{AE}{L}y$ $E=28.3\ast{10}^6psi=1.95\ast{10}^{11}\ N/m^2$

$\frac{\partial F}{\partial y}=\frac{AE}{L}$ and $\frac{W_F}{F}=\frac{\partial F}{\partial y}\frac{W_y}{F}=\frac{W_y}{y}$

$y=100*0.025 mm = 2.5 mm$

$\frac{A}{L}=\frac{F}{Ey}=\frac{1100}{1.95\ast{10}^{11}\ast2.5\ast{10}^3}=2.255\ast{10}^{-6}$

$Considering L=0.1 m => A=\pi . d^2/4 = (0.1)(2.255*10^{-6})$

$d=5.36*10^{-4} m = 0.536 mm$