Question

A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force f, its length increases by l. Another wire of the same material of length 2L and radius 2r, is pulled by a force 2f. Find the increase in length of this wire.

Answer 1

Wire 1

L₁ = L

f₁ = r

A₁ = πr²

F₁ = f

ΔL₁ = 1

Y₁ = Y

Wire 2

L₂ = 2L

f₂ = 2r

A₂ = π(2r)² = 4πr²

F₂ = 2f

ΔL₂ = ?

Y₂ = Y

As Y = `(FL)/(AΔL)` or ΔL = `(FL)/(AY)`

`(F_2L_2)/(A_2Y_2) = (F_2L_2)/(F_1L_1) xx (A_1Y_1)/(A_2Y_2) = (2f2L)/(fL) xx (pir^2 xx Y)/(4pir^2 xx Y)`  

`(F_1L_1)/(A_1Y_1)`

`(ΔL_2)/1 = 4/4 = 1`

ΔL₂ = 1

Thus, the change in the length of the second wire is also the same as that is one.