Two steel wires of the same length, one with radius r and the other with radius 4r, are connected together, end to end, and tied to a wall. An applied force stretches the combination by d = 1.0 mm. How far does the point where the two wires meet move?
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The wire with radius 4r has 16 times the area of the other wire. Therefore, it will be 16 times stiffer (if you think of it as a spring, it will have 16 times the spring constant) than the other wire. Therefore (since each wire must exert the same spring force on the junction, otherwise, the junction would accelerate) the wire with radius 4r will stretch only 1/16 the distance of the other wire.
For each unit distance the wire with radius 4r stretches, the other wire will stretch 16 units. So, the wire with radius 4r stretches 1/17 of the total stretch.
If the wire of radius 4r is attached to the wall, the junction moves 1.0 mm / 17.
If the other wire is attached to the wall, the junction moves 16.0 mm / 17
For each unit distance the wire with radius 4r stretches, the other wire will stretch 16 units. So, the wire with radius 4r stretches 1/17 of the total stretch.
If the wire of radius 4r is attached to the wall, the junction moves 1.0 mm / 17.
If the other wire is attached to the wall, the junction moves 16.0 mm / 17
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So what's the "right" answer? My logic makes enough sense to me (I teach stuff like this for a living) that if someone told me I was wrong, I'd want to know where they think I went wrong, or I'd want to see their work because I suspect that they're actually the one that's wrong.
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