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Two wires A and B of same material have their lengths in the ratio 1 : 2 and their diameters in the ratio 2 : 1. If they are stretched with same force, the ratio of the increase in the length of A to that of B will be
- a)1 : 2
- b)4 : 1
- c)1 : 8
- d)1 : 4
Correct answer is option 'C'. Can you explain this answer?
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Two wires A and B of same material have their lengths in the ratio 1 :...
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Two wires A and B of same material have their lengths in the ratio 1 :...
Given Data:
- Length ratio of wires A and B = 1:2
- Diameter ratio of wires A and B = 2:1
Understanding the Concept:
- When a wire is stretched with a force, it undergoes deformation and increases in length.
- The increase in length of a wire is directly proportional to the force applied and inversely proportional to its cross-sectional area.
- In this case, the force applied to both wires A and B is the same.
Calculating Cross-Sectional Area:
- Let the original length of wire A be x units and its diameter be 2y units.
- The original length of wire B is 2x units and its diameter is y units.
- The cross-sectional area of wire A = π * (2y/2)^2 = π * y^2
- The cross-sectional area of wire B = π * (y/2)^2 = π * (y^2/4)
- The ratio of cross-sectional areas of A and B = (π * y^2) / (π * (y^2/4)) = 4
Calculating Increase in Length:
- Let the increase in length of wire A be Δl units and of wire B be 2Δl units (as per the given length ratio).
- Using the formula Δl = (Force * original length) / (cross-sectional area * Young's modulus),
- The increase in length ratio of A and B = Δl / 2Δl = 1/2 = 1:2
Conclusion:
- The ratio of the increase in the length of wire A to that of wire B is 1:2. Hence, the correct answer is option 'C'.
- Length ratio of wires A and B = 1:2
- Diameter ratio of wires A and B = 2:1
Understanding the Concept:
- When a wire is stretched with a force, it undergoes deformation and increases in length.
- The increase in length of a wire is directly proportional to the force applied and inversely proportional to its cross-sectional area.
- In this case, the force applied to both wires A and B is the same.
Calculating Cross-Sectional Area:
- Let the original length of wire A be x units and its diameter be 2y units.
- The original length of wire B is 2x units and its diameter is y units.
- The cross-sectional area of wire A = π * (2y/2)^2 = π * y^2
- The cross-sectional area of wire B = π * (y/2)^2 = π * (y^2/4)
- The ratio of cross-sectional areas of A and B = (π * y^2) / (π * (y^2/4)) = 4
Calculating Increase in Length:
- Let the increase in length of wire A be Δl units and of wire B be 2Δl units (as per the given length ratio).
- Using the formula Δl = (Force * original length) / (cross-sectional area * Young's modulus),
- The increase in length ratio of A and B = Δl / 2Δl = 1/2 = 1:2
Conclusion:
- The ratio of the increase in the length of wire A to that of wire B is 1:2. Hence, the correct answer is option 'C'.
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Two wires A and B of same material have their lengths in the ratio 1 : 2 and their diameters in the ratio 2 : 1. If they are stretched with same force, the ratio of the increase in the length of A to that of B will bea)1 : 2b)4 : 1c)1 : 8d)1 : 4Correct answer is option 'C'. Can you explain this answer?
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