A small metal bar, whose initial temperature was 30° C, is dropped into a large container of boiling water. How long will it take the bar to reach 70° C if it is known that its temperature increases 2° during the first second? (The boiling temperature for water is 100° C. Round your answer to one decimal place.)

Explanation:

We shall solve this problem on the basis of averaging , otherwise it will require integration which is a complex operation .

According to newton's law of cooling

dQ / dt = k ( T₁ - T₂ )

T₁ is temperature of surrounding  and T₂ is temperature of object .

For the heating by 2 degree

dQ = ms x ΔT , m is mass , s is specific heat and ΔT is rise of

= ms x 2

dt = 1 second

T₁ the average temperature of object = (30 + 32)  / 2 = 31

dQ = ms x ΔT

ms x 2 = k ( 100 - 31 )

k = 2 ms / 69

In the second case bar's temperature rises from 32 to 70

average temperature = 32 + 70 / 2 = 51

If t be the time required

dQ = ms x ( 70 - 32 ) = 38ms

38ms / t = k ( 100 - 51 )

38ms / t = (2ms / 69) x 49

t = 38 x 69 / (2 x 49)

= 26.75 s .  

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