Question

Find the elongation in the rod.

Answer 1

The situation is not clear. If the rod is free to move in horizontal direction, it will move with an acceleration towards right side.

m a = 2 F - F = F.
a = F/m.

In this case there is no elongation. Elongation occurs when the rod is fixed at an end.

If the rod is fixed at one end or two ends, then a tensile force develops inside the rod. It will be equal to the sum of the external forces. This tensile force will be equal at every point along the rod.

If Y is the Young's modulus of the material of the rod and A is the area of cross section,

Elongation = 3F L / (A Y)
using Young's law.

Answer 2

Dear student,

Given :

• length of rod, l
• forces acting on the rod from two opposite sides F and 2F

To find :

• elongation in the rod, Δl

Solution :

∵ elongation, Δl = Fl/ Aγ ____(1)

∴      Δl = (f₁ - f₂)/2 × l/ Aγ

         Δl = [2F - (- F) ]/2 × l/ Aγ

         Δl = 3F l/ 2Aγ

∴ The elongation in the rod is 3F l/ 2Aγ.

Verification :

Let f₁ = f₂ = F . Now,

Then,

        Δl = 2Fl/ Aγ ____(2)

        Δl = Fl/ Aγ    [from (1)]

       

Now, equating (1) and (2), we obtain:

     Fl/ Aγ = ( F₁ + F₂/2) × l/ Aγ    

     Fl/ Aγ = 2F / 2 × l/ Aγ    

      Fl/ Aγ = Fl/ Aγ

∴ Verified!