How much energy does a bar of iron radiate after losing 8.0 × 10^6J?

[physics123]

A bar of iron is 0.5m long, 0.2m wide and 0.1m high (which means its volume is 1.0 × 10−2m3 and its surface area is 3.4 × 10−1m2 ). Iron has a density of 7900 kg m3 , a heat capacity of 400 J kg◦C , and a coefficient of linear expansion of 1.2 × 10−5 . The bar of iron is initially at 600K.

Assuming that the emissivity e = 0.8, what is the rate at which iron radiates energy after it has lost 8.0 × 106J?I know the answer is 220 W, but how do I get that answer? Thanks!

 

[haruspex]

A bar of iron is 0.5m long, 0.2m wide and 0.1m high (which means its volume is 1.0 × 10−2m3 and its surface area is 3.4 × 10−1m2 ). Iron has a density of 7900 kg m3 , a heat capacity of 400 J kg◦C , and a coefficient of linear expansion of 1.2 × 10−5 . The bar of iron is initially at 600K.

Assuming that the emissivity e = 0.8, what is the rate at which iron radiates energy after it has lost 8.0 × 106J?I know the answer is 220 W, but how do I get that answer? Thanks!

A thorough analysis would consider the uneven temperature distribution that would develop within the bar as it cools. It involves Fourier transforms. I shall guess you are not expected to allow for that.
What would be the temperature drop in losing that much energy?

 

[rude man]

Hint: Figure temperature after cooling, then go with Stefan-Boltzmann law, appropriately scaled. I would ignore the expansion coeff. - but, whatever.