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Question

Find the rate of heat flow through a cross section of the rod shown in figure (θ2>θ1). Thermal conductivity of the material of the rod is K.

 


Solution

Tan Φ=r2r1L=yr1x

Differentiating w. r to x 

r2r1=Ldydx0

dydx=r2r1L

r2r1Ldx=dyLr2r1                ...(i)

Qt=kπy2dθdx

Now Qdxt=kπy2dθdx

QLdyt(r2r1)=kπy2Δθ                                                                      [From equation (i)]

dθ=QLdyt(r2r1)kπy2

Integrating both side 

θ2θ1dθ=QLt(r2r1)kπr2r1dyy2

(θ2θ1)=QLt(r2r1)kπ×[1y]r2r1

(θ2θ1)=QLt(r2r1)kπ×[1r11r2]

(θ2θ1)=QLt(r2r1)kπ(r2r1)r2r1

Qt=kπr1r2(θ1θ2)L


Physics
Concepts of Physics
Standard XII

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