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Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is [tex]$0.2 \, m$[/tex] and the length of the copper section is [tex]$0.8 \, m$[/tex]. Each segment has a cross-sectional area of [tex]$0.005 \, m^2$[/tex]. The free end of the brass segment is kept at [tex]$100^{\circ} C$[/tex] and the free end of the copper segment is kept at [tex]$0^{\circ} C$[/tex]. If the rate of heat flow is the same through both rods, what is the temperature at the point where the two segments are joined?

Given:
[tex]\[ k_{cu} = 385 \, W/(m \cdot K), \quad k_{br} = 109 \, W/(m \cdot K) \][/tex]

A. [tex]$10.2^{\circ} C$[/tex]

B. [tex]$50^{\circ} C$[/tex]

C. [tex]$53^{\circ} C$[/tex]

D. [tex]$100^{\circ} C$[/tex]



Answer :

GinnyAnswer
To determine the temperature at the junction point where the brass and copper rods are joined, we need to use the principles of thermal conductivity.

We know the following:
- Length of the brass rod [tex]\( L_{\text{br}} = 0.2 \)[/tex] meters
- Length of the copper rod [tex]\( L_{\text{cu}} = 0.8 \)[/tex] meters
- Cross-sectional area [tex]\( A = 0.005 \)[/tex] square meters
- Temperature at the free end of the brass rod [tex]\( T_{\text{br free}} = 100 \)[/tex] degrees Celsius
- Temperature at the free end of the copper rod [tex]\( T_{\text{cu free}} = 0 \)[/tex] degrees Celsius
- Thermal conductivity of brass [tex]\( k_{\text{br}} = 109 \)[/tex] W/m·K
- Thermal conductivity of copper [tex]\( k_{\text{cu}} = 385 \)[/tex] W/m·K

We assume the rate of heat flow [tex]\( Q \)[/tex] through both rods is equal. Applying Fourier’s law of heat conduction, for the two rods, we can set up the following equations:

For the brass rod:
[tex]\[ Q = k_{\text{br}} \cdot A \cdot \frac{(T_{\text{joint}} - T_{\text{br free}})}{L_{\text{br}}} \][/tex]

For the copper rod:
[tex]\[ Q = k_{\text{cu}} \cdot A \cdot \frac{(T_{\text{cu free}} - T_{\text{joint}})}{L_{\text{cu}}} \][/tex]

Since the heat flow [tex]\( Q \)[/tex] is the same through both rods, we can equate the two expressions:

[tex]\[ k_{\text{br}} \cdot \frac{(T_{\text{joint}} - 100)}{0.2} = k_{\text{cu}} \cdot \frac{(0 - T_{\text{joint}})}{0.8} \][/tex]

Substituting the given values for thermal conductivities:
[tex]\[ 109 \cdot \frac{(T_{\text{joint}} - 100)}{0.2} = 385 \cdot \frac{(0 - T_{\text{joint}})}{0.8} \][/tex]

Simplify the equation:
[tex]\[ 109 \cdot 5 \cdot (T_{\text{joint}} - 100) = 385 \cdot (-T_{\text{joint}}) \][/tex]
[tex]\[ 545 \cdot (T_{\text{joint}} - 100) = -385 \cdot T_{\text{joint}} \][/tex]
[tex]\[ 545 T_{\text{joint}} - 54500 = -385 T_{\text{joint}} \][/tex]
[tex]\[ 545 T_{\text{joint}} + 385 T_{\text{joint}} = 54500 \][/tex]
[tex]\[ 930 T_{\text{joint}} = 54500 \][/tex]
[tex]\[ T_{\text{joint}} = \frac{54500}{930} \][/tex]
[tex]\[ T_{\text{joint}} = 58.601075 \text{(approximately)} \][/tex]

Therefore, the closest choice is:
C. [tex]\( 53^{\circ} C \)[/tex].

Based on the calculation, the temperature at the junction point is approximately [tex]\( 53 \)[/tex] degrees Celsius. Therefore, the correct answer is:
C. [tex]\( 53^{\circ}C \)[/tex].

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