Consider a 100 m long wire of density `7700 kg //m^(3),` Young's modulus of 39 GPa and radius 10 cm. This wire is used in an elevator to lift a load of mass 5000 kg and then the elongation of wire is fount to be `( K xx 10 ^(-3))/(pi)m.` Then what is K=?

Answer

`43.85`

Explanation

Weight of wire of elevator,
`W = V xxrho xx g ......(because m = V _(rho ))`
`= Ahpg ...(because V = A xx h )`
`= pi r ^(2) h rho g `
`= (22)/(7) xx(0.1) ^(2) xx 100 xx 7700 xx 1 0 = 2. 42 xx 10 ^(5) N`
Stretching force,
`F _(1) = (w)/(2) = ( 2. 42 xx 10 ^(5))/(2) = 1. 21 xx 10 ^(5) N`
Force due to weight of cabin,
`F _(2) = mg = 5000 xx 10 = 50, 000N`
Total force `F = F _(1) + F _(2)`
`= 1. 21 xx 10 ^(5) + 0.5 xx 10 ^(5)`
`= 1. 71 xx 10 ^(5) N`
Extension in wire
` l = ( FL )/( AY ) = ( 1. 71 xx 10 ^(5) xx 100)/( pi xx ( 0.1) ^(2) xx 39 xx 10 ^(9)) = ( 43 .85 xx 10 ^(-3))/( pi ) m `
`implies C = 43.85`