Question

The ends A and B of a rod of length 10cm long have their temperature distribution kept at 20 deg C and 70 deg C. Find the steady state temperature distribution of the rod. *

1 point

u(x)=5x+20

u(X)=2X+50

u(x)=5x+2

u(X)=2X+5​

Answer 1

SOLUTION

GIVEN

The ends A and B of a rod of length 10cm long have their temperature distribution kept at 20° C and 70° C.

TO DETERMINE

The steady state temperature distribution of the rod. 

• u(x)=5x+20

• u(X)=2X+50

• u(x)=5x+2

• u(X)=2X+5

EVALUATION

The equation for heat flow is given by

$\displaystyle \sf{ \frac{ \partial u}{ \partial t} = { \alpha }^{2} \: \frac{ {\partial}^{2} u}{ \partial {x}^{2} } \: \: }$

In steady state temperature the above equation reduces to

$\displaystyle \sf{ \frac{ {\partial}^{2} u}{ \partial {x}^{2} } = 0 \: \: }$

The solution is given by

$\sf{u = ax + b \: \: } \: \: ....(1)$

where a and b constants

Now it is given that

At x = 0, u = 20 and at x = 10, x = 70

Using x = 0, u = 20 we get from (1)

$\sf{20 = 0 + b}$

$\implies \sf{b = 20 \: }$

Again Using x = 10, u = 70 we get from (1)

$\sf{70 = 10a + b \: }$

$\implies \sf{70 = 10a + 20 \: }$

$\implies \sf{10a = 50}$

$\implies \sf{a = 5}$

From Equation (1) we get by putting the value of a and b

$\sf{u = 5x + 20 \: }$

FINAL ANSWER

In steady state temperature distribution of the rod

$\boxed{ \sf{ \: \: u (x)= 5x + 20 \: \: \: }}$

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

Divergence of r / r^3 is

(a) zero at the origin

(b) zero everywhere

(c) zero everywhere except the origin

(d) nonzero

https://brainly.in/question/22316220