A certain wire, 3 m long, stretches by 1.2 mm when under tension 200 N. Does an equally thick wire 6 m long, made of the same material and under the same tension, stretch by __ ?. A wire with twice the diameter, 3 m long, made of the same material and under the same tension, stretches by what amount?
Question
A certain wire, 3 m long, stretches by 1.2 mm when under tension 200 N. Does an equally thick wire 6 m long, made of the same material and under the same tension, stretch by __ ?. A wire with twice the diameter, 3 m long, made of the same material and under the same tension, stretches by what amount?
Solution 1
The stretch of a wire under tension is directly proportional to its length and inversely proportional to its cross-sectional area.
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For the first part of the question, we are asked to find the stretch of a wire that is twice as long as the original wire, but with the same diameter and under the same tension. Since the stretch is directly proportional to the length, a wire that is twice as long will stretch twice as much. Therefore, a 6 m long wire will stretch by 2 * 1.2 mm = 2.4 mm.
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For the second part of the question, we are asked to find the stretch of a wire that is the same length as the original wire, but with twice the diameter and under the same tension. The cross-sectional area of a wire is proportional to the square of its diameter, so a wire with twice the diameter has four times the cross-sectional area. Since the stretch is inversely proportional to the cross-sectional area, a wire with four times the cross-sectional area will stretch one-fourth as much. Therefore, a wire with twice the diameter will stretch by 1.2 mm / 4 = 0.3 mm.
Solution 2
The stretch of a wire under tension is directly proportional to its length and inversely proportional to its cross-sectional area.
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For the first part of the question, we are asked to find the stretch of a wire that is twice as long as the original wire, but with the same diameter and under the same tension. Since the stretch is directly proportional to the length, a wire that is twice as long will stretch twice as much. Therefore, a 6 m long wire will stretch by 2 * 1.2 mm = 2.4 mm.
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For the second part of the question, we are asked to find the stretch of a wire that is the same length as the original wire, but with twice the diameter and under the same tension. The cross-sectional area of a wire is proportional to the square of its diameter, so a wire with twice the diameter has four times the cross-sectional area. Since the stretch is inversely proportional to the cross-sectional area, a wire with four times the cross-sectional area will stretch one-fourth as much. Therefore, a wire with twice the diameter will stretch by 1.2 mm / 4 = 0.3 mm.
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