A body of mass 3.14 kg is suspended from one end of a wire of length 10.0 m. The radius of the wire i changing uniformly from `9.8 xx 10^(-4)` mat one end to `5.0xx10^(-4)` m at the other end. Find the change in the lengt of the wire in mm. Young's modulus of the material of wire is `2 xx 10^(11) Nm^(-2)`

Answer

1

Explanation

Figure shows a wire PQ of length L having radius `r_(1)` at end P increasing uniformly to a radius `r_(2)` at end Q. Consider an element AB of the wire of length dy at a distance y from end P. Let r be the radius of the wire at y and ( r+dr) at (y+dy) . As the radius increases uniformly from P to Q .
`(dr)/(dy) ` = constant say C
Where `C = (r_(2) -r_(1))/(L) ` Thus dy `= (dr)/(C )`

Let us first fint the extension dl produced in teh element of length dy due to a force F= mg . Sress in the element `= (F)/(pi r^(2)) ` and strain in it `= (dl)/(dy) ` .Hence ` y= ("stress ")/("strain") = ((F)/(pir^(2)))/((dl)/(dy))=(Fdy)/(pi r^(2) dl)`