A slightly tapring wire of length L and end radii 'a' and 'b' is subjected to stretching forces F, F as shown in

If Y is the Young's Modulus, calculate the extension prouduced in the wire.

Explanation

Consider a small element cdef of wire of length dx, radius r at a distance x from the end B. The gradient in radius k is constant and is given by `k = (b-a)/(L) …. (i) ` If r is the radius at cd and (r + dr) at fe, the `k = ((r + dr) -r)/((x + dx) - x) = (dr)/(dx) or dx = dr//k....(ii)` If dl is the extension produced in the wire is given by `I = int_a^b (F)/(Y pi) xx (dr)/(k r^2) = (-F)/(Y pi k) [ (1)/(r )]_a^b = (F)/(Y pi k) [(1)/(a) - (1)/(b)] = (F(b-a))/(Y pi kxxab) ["using "(i)]`
` =(FL)/(Y pi ab)`