A copper wire is stretched at a temperature of `150^(@)C` between two strong fixed walls. At what temperature, will the wire break when it cools ? Assume Hooke's law is applicable till the wire breaks.
`alpha_(Cu) = 1.6 xx 10^(-5)`/degree
`Y_("copper") = 11.8 xx 10^(10) N//m^(2)`
Ultimate strength of copper `= 2.45 xx 10^(8) N//m^(2)`

Options

A. `5^(@)C`
B. `10^(@)C`
C. `20^(@)C`
D. `130^(@)C`

Answer

C

Explanation

When the temperature falls to `theta^(@)C`,
Change of length `= l alpha (150 - theta)`
`:.` Strain `= (l alpha (150 - theta))/(l) = alpha (150 - theta)`
and stress `= Y alpha (150 - theta)`
This must be the ultimate strength of copper.
`Y = alpha (150 - theta) = 2.45 xx 10^(8)`
`rArr 11.8 xx 10^(10) xx 1.6 xx 10^(-5) (150 - theta)`
`= 2.45 xx 10^(8)`
`rArr (150 - theta) = (2.45 xx 10^(8))/(11.8 xx 1.6 xx 10^(8))`
`= (2.45 xx 10^(3))/(11.8 xx 1.6) = (2.45 xx 10^(3))/(18.88) = 130`
`:. theta = 20^(@)C`