JEE Main 2021 · 25 July · morning · Single correct · +4 / −1
Two wires of same length and radius are joined end to end and loaded. The Young's modulii of the materials of the two wires are Y₁ and Y₂. The combination behaves as a single wire then its Young's modulus is :
- (a) Y = 2Y₁Y₂/3(Y₁ + Y₂)
- (b) Y = 2Y₁Y₂/(Y₁ + Y₂)
- (c) Y = Y₁Y₂/2(Y₁ + Y₂)
- (d) Y = Y₁Y₂/(Y₁ + Y₂)
Answer and solution
Answer (b) Y = 2Y₁Y₂/(Y₁ + Y₂)
Solution
Physics sir · 3 steps
Same force goes through both wires, so add the two extensions and remember the total length is 2L.
- Same force, same area. The wires are in series, so the same load F stretches each one. Each has length L and cross-section A, so the extensions are Δ L₁ = FL/AY₁ and Δ L₂ = FL/AY₂.
- Treat as one wire. The combined wire has length 2L, the same area A and the same force F. Its extension is the sum Δ L₁ + Δ L₂, so Y = (F × 2L)/A(Δ L₁ + Δ L₂).
Y = (F × 2L)/A(FL/AY₁ + FL/AY₂) = 2/(1/Y₁ + 1/Y₂) = 2Y₁Y₂/(Y₁ + Y₂)
- So. F, A and L all cancel, and the combination behaves like a wire with Y = 2Y₁Y₂/(Y₁ + Y₂), which is option (b).
The idea to keep. For wires joined end to end, the force is common and the extensions add. Always use the total length of the combination when you write the new Young's modulus.
Answer from the NTA final answer key. Solution by PrepWiser.