| DIRECTION: Read the passage given below and answer the questions that follows: |
| In a thermally insulated tube of cross sectional area \[\frac{GMm}{2R}\] a liquid of thermal expansion coefficient \[{{10}^{-3}}\,{{K}^{-1}}\] is flowing. Its velocity at the entrance is \[0.1\text{ }m/s\]. At the middle of the tube a heater of a power of 10kW is heating the liquid. The specific heat capacity of the liquid is 1.5 kJ/(kg K), and its density is \[1500\text{ }kg/{{m}^{3}}\] at the entrance. |
A) 1450 kg/m3
B) 1400 kg/m3
C) 1350 kg/m3
D) None of these
Correct Answer: C
Solution :
| \[{{\text{ }\!\!\rho\!\!\text{ }}_{1}}{{v}_{1}}{{A}_{1}}={{\text{ }\!\!\rho\!\!\text{ }}_{2}}{{v}_{2}}{{A}_{2}}\] |
| \[\text{m =1500 kg/}{{\text{m}}^{\text{3}}}\text{ }\!\!\times\!\!\text{ 0}\text{.1m/s }\!\!\times\!\!\text{ 4}{{\left( \text{cm} \right)}^{\text{2}}}\] |
| \[ms\Delta T=10000\] |
| \[1500\times 0.1\times 4\times {{10}^{-4}}\times 1500\times \Delta T=10000\] |
| \[\Delta T=\frac{10000}{90}=\frac{1000}{9}{}^\circ C\] |
| \[{{\text{ }\!\!\rho\!\!\text{ }}_{\text{2}}}=\frac{{{\text{ }\!\!\rho\!\!\text{ }}_{\text{1}}}}{\left( 1+\text{ }\!\!\gamma\!\!\text{ }\Delta T \right)}=\frac{1500}{\left( 1+1\times {{10}^{3}}\times \frac{1000}{9} \right)}=1350kg/{{m}^{3}}\]\[{{\text{ }\!\!\rho\!\!\text{ }}_{2}}{{v}_{2}}{{A}_{2}}={{\text{ }\!\!\rho\!\!\text{ }}_{1}}{{v}_{1}}{{A}_{1}}\] |
| \[\Rightarrow \,1350\times {{v}_{2}}=1500\times 0.1\] |
| \[{{v}_{2}}=1/9m/s\] |
| \[\therefore \]Volume rate of flow at the end of tube |
| \[={{A}_{2}}{{v}_{2}}=4\times {{10}^{-4}}\times \frac{1}{9}\] |
| \[=\frac{4}{9}\times {{10}^{-4}}{{m}^{3}}=\frac{40}{9}\times {{10}^{-5}}{{m}^{3}}\] |
| Volume rate of flow at the entrance = \[{{A}_{1}}{{v}_{1}}\] |
| \[=0.1\times 4\times {{10}^{-4}}=4\times {{10}^{-5}}{{m}^{3}}\] |
| Hence, difference of volume rate of flow at the two ends |
| \[=\left( \frac{40}{9}-4 \right)\times {{10}^{-5}}=\frac{4}{9}\times {{10}^{-5}}{{m}^{3}}\] |
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