A theoretical model suggests that the power P generated by a wind turbine depends on the blade length L , wind speed v , air density , and the acceleration due to gravity g . If the relationship is given by P L^ v^ ^ g^ , which of the following sets of exponents is dimensionally consistent?

Step-by-step solution

Write the dimensional formula for each physical quantity: Power P = [M L^2 T⁻³] Length L = [L] Velocity v = [L T⁻¹] Density = [M L⁻³] Acceleration due to gravity g = [L T⁻²] Using the principle of dimensional homogeneity: [M L^2 T⁻³] = [L]^ [L T⁻¹]^ [M L⁻³]^ [L T⁻²]^ [M L^2 T⁻³] = [M^ L^ + - 3 + T^ - - 2 ] Equating the powers of M , L , and T on both sides: For M : = 1 For T : - - 2 = -3 + 2 = 3 For L : + - 3 + = 2 Substitute = 1 into the L equation: + - 3(1) + = 2 + + = 5 Now check the given options to see which s