Question

A given solid rectangular bar (mass = M) has an area of cross section A and a length L. Its density ρ , and its specific heat capacity, s, vary with distance, x from one end, as

ρ(x) = k₁x 

and s(x) = k₂x  , respectively.

The heat needed to raise the temperature of this bar by an amount, θ , is

(1) 1/2 k₂ ML³

(2) 2/3 k₂ ML

(3) 1/3 k₂ML³

(4) 1/2 k₂ ML²

Answer 1

(2) 2/3 k₂ ML

The mass, dm, of a segment of the bar, of length dx, is

The specific heat capacity, of this segment, is

s(X) = k₁x

Hence the heat needed to raise its temperature, by an amount θ , is

Hence the total heat needed is