Question 971
CBSE Class 11 Physics · Work, Energy and Power · Work and kinetic energy
A wind turbine intercepts area A in air of density rho moving normally at speed v. Find (a) mass crossing in time t(b) its kinetic energy(c) electrical power for 25% efficiency when \(A=30\,\mathrm{m^{2}}\), \(v=36\,\mathrm{km\,h^{-1}}\) and rho=\(1.2\,\mathrm{kg\,m^{-3}}\).
Solution to Question 971
Complete workingPrinciple. The swept air volume is Avt. Wind power is kinetic energy per time, and electrical power is the efficiency times this rate.
\[m=\rho Avt\]
\[\begin{aligned}K &= \tfrac12mv^2\\&= \tfrac12\rho Av^3t\end{aligned}\]
\[\begin{aligned}v &= \frac{36}{3.6}\\&= 10.0\ \mathrm{m\,s^{-1}}\end{aligned}\]
\[\begin{aligned}P_e &= 0.25\left(\tfrac12\rho Av^3\right)\\&= 0.25(0.5)(1.2)(30)(10^3)\\&= 4.50\times10^3\ \mathrm W\end{aligned}\]
Final answers by part
(a)
\[m=\rho Avt
(b)
\ K=\tfrac12\rho Av^3t
(c)
\ P_e=4.50\ \mathrm{kW}.\]