Two wires made up of the same material are subjected to forces in the ratio of 1:4 Their lengths - Physics - Mechanical Properties Of Solids

Question

Two wires made up of the same material are subjected to forces
in the ratio of 1:4 Their lengths are in the ratio of 8:1 and
diameter in the ratio of 2:1 Find the ratio of their extensions

Rahul Verma · Accepted Answer

Since both the wires are of same material there modulus of
Elasticity is same E=StressStrainStress=ForceArea Strain=change in
length(Extension)original lengthLet force for first wire is F
Then force for second wire is 4F Diameters are in 2:1 ratio
⇒Areas are in 4:1 ratio (∵Area is directly proportional
to d2)
Area of first wire is 4A
Area of second wire is A
Lengths are in Ratio 8:1 Length of first wire is 8l Length of
second wire is l
let the extension in first wire be ∆l1 let the extension in
second wire be ∆l2
E for first wire=
(F4A)(∆l18l)=F×8l4A×∆l1
E for second wire=
(4FA)(∆l2l)=4F×lA×∆l2
We know:E for first wire = E for second wire

⇒F×8l4A×∆l1=4F×lA×∆l2
⇒∆l1∆l2=12 or 1:2

Two wires made up of the same material are subjected to forces in the ratio of 1:4. Their lengths are in the ratio of 8:1 and diameter in the ratio of 2:1. Find the ratio of their extensions.