A wire Y of cross-sectional area A and length l is joined to a second wire Z of cross-sectional area 2A and length 2l. Wire Z is fixed at one end and a force F is applied to the ot... A wire Y of cross-sectional area A and length l is joined to a second wire Z of cross-sectional area 2A and length 2l. Wire Z is fixed at one end and a force F is applied to the other end of wire Y. The wires are made of the same material. Wire Y extends by a distance x. What is the extension of wire Z?
Understand the Problem
The problem describes two wires, Y and Z, made of the same material but with different cross-sectional areas and lengths, connected in series. A force F is applied to wire Y, causing it to extend by a distance x. The question asks us to determine the extension of wire Z, given the dimensions and material properties of both wires.
Answer
The extension of wire Z is x.
The extension of wire Z is x, as both wires experience the same stress due to the equal force applied and are made of the same material, thus having the same strain.
Answer for screen readers
The extension of wire Z is x, as both wires experience the same stress due to the equal force applied and are made of the same material, thus having the same strain.
More Information
Since both wires are made of the same material, they have the same Young's modulus. The extension of a wire is given by (x = \frac{FL}{AY}), where F is the force, L is the length, A is the cross-sectional area, and Y is Young's modulus. For wire Y, the extension is (x = \frac{Fl}{AY}). For wire Z, the extension is (x' = \frac{F(2l)}{(2A)Y} = \frac{Fl}{AY}). Thus, (x' = x).
Tips
A common mistake is to only consider the change in length and area independently, without considering that both wires are made of the same material and the same force is applied to both wires, both wires experience the same stress.
Sources
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