there are two wires of same material.their radii and length are both in the ratio 1:2.if the extensions produced are equal, what is the ratio of the load ?
ans:
1)1:2
Y=lxF1/Axe=2lxF2/4Axe.
Here, e is same and after cancellation, F1/F2=1/2
Therefore, the ratio is 1:2.
To find the ratio of the load, we can use the concept of stress and strain. Stress is defined as the force applied per unit area and strain is defined as the extension produced per unit length.
Given that the two wires are made of the same material, we can assume that the stress is the same for both wires since they are subjected to the same load.
Let's denote the radii of the wires as r and 2r, and the lengths as L and 2L respectively, where the ratio of radii and lengths is 1:2.
The extension produced in a wire is directly proportional to the applied load and inversely proportional to the cross-sectional area. Mathematically, it can be represented as:
strain = extension / original length
Since the extensions produced are equal, let's say it is 'e'. Thus, for the wire with radius r, the strain is e/L, and for the wire with radius 2r, the strain is e/2L.
Since the stress is the same for both wires, we can equate the stress-strain ratio:
Stress1/Strain1 = Stress2/Strain2
The stress is the force applied divided by the cross-sectional area. Let's denote the force applied as 'F' and the Young's modulus of the material as 'Y'.
For the wire with radius r:
Stress1 = F / (πr^2)
For the wire with radius 2r:
Stress2 = F / (4πr^2)
Substituting these values into the stress-strain ratio equation:
(F / (πr^2)) / (e/L) = (F / (4πr^2)) / (e/2L)
Simplifying it further:
2 / 1 = 1 / 4
This implies that the load ratio is 1:2.
Therefore, the ratio of the load for the two wires is 1:2.