Review. A steel wire and a copper wire, each of diameter 2.000 mm, are joined end to end. At 40.0°C, each has an unstretched length of 2.000 m. The wires are connected between two fixed supports 4.000 m apart on a tabletop. The steel wire extends from x = –2.000 m to x = 0, the copper wire extends from x = 0 to x = 2.000 m, and the tension is negligible. The temperature is then lowered to 20.0°C. Assume the average coefficient of linear expansion of steel is 11.0 × 10–6 (°C)–1 and that of copper is 17.0 × 10–6 (°C)–1. Take Youngs modulus for steel to be 20.0 × 1010 N/m2 and that for copper to be 11.0 × 1010 N/m2. At this lower temperature, find (a) the tension in the wire and (b) the x coordinate of the junction between the wires.
(a)
The tension in the wire.
Answer to Problem 19.72CP
The tension in the wire is
Explanation of Solution
Given Info: The diameter of both the wires is
Formula to calculate the radius of the wire is,
Here,
Substitute
Thus, the value of the radius is
The initial area of cross section of the steel wire is,
Substitute
Thus, the value of the initial area of cross section of the steel wire is
Substitute
Thus, the value of the initial area of cross section of the copper wire is
When the wire is stretched its length and its area of cross section both have changed.
Formula to calculate the new area of cross section of the steel wire is,
Substitute
Thus, the value of the final area of cross section of the steel wire is
Formula to calculate the new area of cross section of the copper wire is,
Substitute
Thus, the value of the final area of cross section of the copper wire is
Formula to calculate the final length of the steel wire under a tension
Here,
Formula to calculate the final length of the copper wire under a tension
Here,
Formula to calculate the tension in the composite wire is,
Substitute
Conclusion:
Thus, the tension in the wire is
(b)
The x-coordinate of the junction between the wires.
Answer to Problem 19.72CP
The final x-coordinate is
Explanation of Solution
Given Info: The diameter of both the wires is
Formula to calculate the final length of the steel wire under a tension
Here,
Substitute
Thus, the final length of the steel wire under a tension
Formula to find final x coordinate is,
Here,
Substitute
Conclusion:
Therefore, the final x-coordinate is
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Chapter 19 Solutions
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