A) \[n\text{ }-\text{ }times~\]
B) \[{{n}^{2}}-times~\]
C) \[{{n}^{3}}-times~\]
D) \[{{n}^{4}}-times~\]
Correct Answer: C
Solution :
If the motor pumps water (density\[=\rho \]) continuously through a pipe of area of cross- section A with velocity v, then mass flowing out per second. \[m=Av\rho \] ...(i) Rate of increase of kinetic energy \[=\frac{1}{2}m{{v}^{2}}=\frac{1}{2}(Av\rho ){{v}^{2}}\] ?(ii) Mass m, flowing out per sec, can be increased to m' by increasing v to v? then power increases from P to P'. \[\frac{P'}{P}=\frac{\frac{1}{2}A\rho {{v}^{,3}}}{\frac{1}{2}A\rho {{v}^{3}}}\]or \[\frac{P'}{P}={{\left( \frac{v'}{v} \right)}^{3}}\] Now, \[\frac{m'}{m}=\frac{A\rho v'}{A\rho v}=\frac{v'}{v}\] As \[m'=nm,\,\,v'=nv\] \[\therefore \] \[\frac{P'}{P}={{n}^{3}}\]\[\Rightarrow \] \[P'={{n}^{3}}P\]
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