An thin rod of negligible mass and area of cross - section `4xx10^(-6)m^(2)`, suspended vertically from one end, has a length of 0.5 m at `100^(@)C`. The rod is cooled to `0^(@)C` but prevented from contracting by attaching a mass at the lower end. The value of this mass is ( in Kg )
(Given, coefficient of linear expansion is `10^(-5^(@))C^(-1)`, Young's modulus is `Y=10^(11)Nm^(-2) and g=10ms^(-2))`

Answer

40

Explanation

The change in length due to decrease in temperature
`Deltal_(1) = L alpha Delta theta `
`= (0.5) ( 10^(-5)) (0-100)`
`Deltal_(1) = - 0.5 xx 10^(-3) m`
Negative sign implies that length is decreasing. Now let M be the mass attached to the lower end. Then, change in length due to suspension of load is
`Deltal_(2) ((Mg)L)/(AY) = ((M) (10) (0.5))/((4xx 10^(-6)) (10^(1)))`
`Delta l_(2) = (1.25 xx 10^(-5) ) M ` ...... (ii)
`Delta_(1) + Delta l_(2) =0`
`implies ( 1.25 xx 10^(-5)) M = (0.5 xx 10^(-3))`
( from equations (i) and (ii))
`implies M = (0.5 xx 10^(-3))/(1.25 xx 10^(-5)) ` Kg
`implies M = 40 kg`