A piece of modeling clay is rolled into a cylinder and its resistance R is measured with an ohmmeter.  The clay is then rolled out so it is twice as long.  What is the new resistance measured...

A piece of modeling clay is rolled into a cylinder and its resistance R is measured with an ohmmeter.  The clay is then rolled out so it is twice as long.  What is the new resistance measured (considering how the cross-section area changes)?

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Let volume of the cay  be V cubic unit, length be initial be L and radius circular of cross section be  r.Thus

we have

`V=pi r^2 L`        (i)

when legth of cylinder doubled i.e 2L ,since volume remain same so radius of circular cross section will reduced to   `r_1` (say). So volume will be

`V=pir^2_1(2L)`

`V=2pir^2_1L `               (ii)                 

From (i) and (ii)  we have  

`r^2=2r^2_1`                 (iii)

Thus

`pir^2=pi(2r^2_1)`

we know

`R=rhoL/A`

`R=rhoL/(pir^2)`                  (iv)

let `R_1` be the resistance when legth 2L

`R_1=rho(2L)/(pir^2_1)`                 , usin (iii) ,we will get

`R_1=rho(2L)/(pir^2/2)`

`R_1=rho(4L)/(pir^2)`                     (v)

From (iv) and (v) ,we have

`R_1=4R`

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