Question

Wind Turbine As of January 2018 , the world's tallest wind turbine was located in Gaildorf, Germany, with a hub height of 178 meters and a rotor diameter of 137 meters. If the blades turn at a rate of 14 revolutions per minute, what is the linear speed of the blade tip, in \(\mathrm{km} / \mathrm{h} ?\)

Short answer

The linear speed of the blade tip is approximately 361.6 km/h.

Step-by-step solution

Calculate the circumference of the rotor

First, find the circumference of the rotor. The rotor diameter is given as 137 meters. The circumference (\text{C}) of a circle is given by the formula \( C = \pi d \) where \( d \) is the diameter. So, \( C = \pi \times 137 \approx 430.56 \text{ meters} \).

Calculate the distance traveled by the blade tip in one minute

The blades turn at a rate of 14 revolutions per minute (RPM). To find the distance traveled by the blade tip in one minute, multiply the circumference by the number of revolutions per minute: \( D = 430.56 \times 14 \approx 6027.84 \text{ meters} \).

Convert the distance to kilometers

Convert the distance from meters to kilometers. Since there are 1000 meters in one kilometer, \( D = \frac{6027.84}{1000} \approx 6.03 \text{ kilometers} \).

Calculate the linear speed in kilometers per hour

The previous result is the distance traveled in one minute. To find the distance per hour, multiply by 60 (since there are 60 minutes in an hour). So, the linear speed of the blade tip is \( V = 6.03 \times 60 \approx 361.6 \text{ kilometers per hour} \).

Key concepts

Circumference Calculation

Let's start by understanding the concept of circumference. The circumference of a circle is the distance around it. Think of it as the perimeter of the circle. This is important because it tells us how far the blade of the wind turbine travels in one full revolution.
To calculate the circumference, we use the formula: \( C = \pi d \). Here, \( d \) stands for the diameter of the circle. So, if the diameter of our wind turbine rotor is 137 meters, the circumference would be \( C = \pi \times 137 \approx 430.56 \) meters.
This means that each time the blade completes one full turn, it covers 430.56 meters! This concept will be pivotal as we proceed to calculate the linear speed of the blade tip.

Revolutions Per Minute (RPM)

Next, we need to understand the term 'Revolutions Per Minute', or RPM. RPM measures how many times an object (like our wind turbine blade) completes one full circle in one minute.
In our case, the wind turbine blades turn at a rate of 14 RPM. This means every minute, the blade makes 14 complete circles. The distance traveled in one minute can be calculated by multiplying the RPM by the circumference we found earlier.
So, we do \( 430.56 \times 14 \approx 6027.84 \) meters per minute. This shows how far the blade tip moves in just one minute!

Unit Conversion

Finally, let's dive into unit conversion. It’s important to convert units to make sure our final answer is in the required format.
In this exercise, we need the linear speed in kilometers per hour (km/h). First, we found the distance traveled in one minute, which was 6027.84 meters.
To convert this to kilometers, we divide by 1000, giving us 6.03 kilometers per minute.
Then, because there are 60 minutes in an hour, we multiply by 60 to convert our answer to kilometers per hour: \( 6.03 \times 60 \approx 361.6 \) kilometers per hour.
And there we have it! This is the linear speed of the blade tip: 361.6 km/h. Now you can see how unit conversion plays a key role in this kind of problem-solving.