Windmill Blade Tip Speed and Acceleration Calculation

How can we calculate the blade tip speed as a fraction of the speed of sound?

a) The blade tip speed is what fraction of the speed of sound?

What is the formula to calculate the acceleration of the blade tips?

b) What is the acceleration of the blade tips, in m/s²?

Final answer:

The blade tip speed is 11.6% of the speed of sound, and the acceleration of the blade tips is 1360.44 m/s².

a) To find the blade tip speed as a fraction of the speed of sound, we need to calculate the actual tip speed and then divide it by the speed of sound. The actual tip speed is given by the formula: tip speed = length of blade × number of revolutions per minute. Using the given values, the tip speed is 43.0 meters × 55.0 revs/min = 2365 meters/min.

However, we need to convert the tip speed to meters per second to compare it with the speed of sound. There are 60 seconds in a minute, so the tip speed is 2365 meters/min ÷ 60 min/sec = 39.42 meters/sec.

Now, we can calculate the fraction of the speed of sound by dividing the tip speed by the speed of sound: fraction = tip speed ÷ speed of sound = 39.42 meters/sec ÷ 340 meters/sec = 0.116 or 11.6%.

b) To find the acceleration of the blade tips, we can use the formula for centripetal acceleration: acceleration = (angular velocity)^2 × radius. The angular velocity is the number of revolutions per minute converted to radians per second, which is given by: angular velocity = (2π radians/rev) × (55 revs/min) ÷ (60 sec/min) = 5.75 radians/sec.

Now, we can calculate the acceleration by multiplying the square of the angular velocity by the radius: acceleration = (5.75 radians/sec)^2 × 43.0 meters = 1360.44 meters/sec².