A dopey resistance question.

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You have a length of bare resistance wire a fixed contact and a sliding contact.Using these things only how can you make a variable resistor such that when the sliding contact is moved along the wire the resistance varies smoothly from zero to a maximum and then to zero?You are not allowed to cut the wire.
 

Gokul43201

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Form the wire (of length L and resistance R) into a closed loop. If the sliding contact is at a distance d from the fixed contact, the resistance will be the equivalent parallel resistance from segments of length d and L-d, which should be:

R_eff = [R(d/L)*R((L-d)/L)]/[R(d/L)+R((L-d)/L)] = Rd(L-d)/(L^2),

and as d increases from 0, d(L-d) increases smoothly from 0 till d=L/2 (where R_eff is at its maximal value of R/4) and decreases smoothly to 0 after that (this part maps onto the common puzzle of maximizing the area of a rectangle of given perimeter, but if you don't like that, then the first derivative is L-2d, and the second is -2).
 
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Well done Gokul 43201.
 

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