A scientific model of a wind turbine's power output P is described by the equation P = \fr… — ACT

A scientific model of a wind turbine's power output $P$ is described by the equation $P = \frac{1}{2} \rho A v^3 \eta$ , where $\rho$ is the air density, $A$ is the swept area of the rotor blades ( $A = \pi r^2$ , where $r$ is the blade length), $v$ is the wind velocity, and $\eta$ is the turbine efficiency. Match each modification to the turbine's operating conditions or physical dimensions (on the left) with its corresponding mathematical effect on the power output $P$ (on the right), assuming all other variables remain constant.

The wind velocity $v$ is doubled.The power output $P$ increases by a factor of 8.
The blade length $r$ is doubled.The power output $P$ increases by a factor of 4.
The air density $\rho$ is quadrupled and the wind velocity $v$ is halved.The power output $P$ decreases by a factor of 2 (is halved).
The turbine efficiency $\eta$ is halved, the air density $\rho$ is halved, and the blade length $r$ is halved.The power output $P$ decreases by a factor of 16.

Answer

The correct matches are: doubling wind velocity increases power by a factor of 8; doubling blade length increases power by a factor of 4; quadrupling air density while halving wind velocity halves the power; halving efficiency, air density, and blade length decreases power by a factor of 16.

Each physical modification is correctly matched to its mathematical effect by evaluating the mathematical proportion of each variable in the power equation:

P \propto \rho \cdot r^2 \cdot v^3 \cdot \eta

Step-by-Step Solution

Analyze the relationship between power

P

and wind velocity

v

using

P \propto v^3

Doubling

v

scales

P

2^3 = 8

Power is proportional to the cube of wind velocity.

Analyze the relationship between power

P

and blade length

r

using

P \propto A

and

A = \pi r^2

Doubling

r

scales the swept area

A

2^2 = 4

, which scales

P

by 4.

Power is directly proportional to swept area, which scales with the square of the blade length.

Analyze the combined effect of quadrupling air density

\rho

and halving wind velocity

v

The scaling factor is

4 \times (0.5)^3 = 0.5

, halving the power.

Power is proportional to density and velocity cubed, so the scaling factors multiply.

Analyze the combined effect of halving efficiency

\eta

, halving density

\rho

, and halving blade length

r

The scaling factor is

0.5 \times 0.5 \times (0.5)^2 = 0.0625 = \frac{1}{16}

, decreasing power by a factor of 16.

The proportional changes of the independent variables scale the overall expression, taking into account the square on the blade length.

Key Concept

Analyzing proportional and power-law relationships in scientific equations.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic