How much faster(in miles per hour) is the tip of the blade than the base of the blade?

Question

Full problem: It takes 4.5 second a for a windmill to make a rotation. Each blade extends 116 feet from the central hub that is 6 feet in diameter. How much faster(in miles per hour) is the tip of the blade moving than the base of the blade? 

Answer 1

Kailey,

 

We can solve this by determine the circumference of the circles made by the tip of the blade and the base of the blade and then dividing by time to get the speed in feet per second (then convert to miles per hour, i.e.  

 

v =(pi·d)/t, where t = 4.5 s

 

For the hub this is easy, the diameter is simply 6 feet.

 

For the blades the diameter is going to be the length of the blade x 2, plus the diameter of the hub, so the diameter at the tip of the blades is:  d_blade = (232 + 6) feet = 238 feet.

 

v_tip  = [pi·(238)/4.5] ft/s = 166.2 ft/s

 

v_base = [pi·6]/4.5 ft/s = 4.189 ft/s

 

The difference in speeds is thing Δv =  v_tip - v_base  = 161.9 ft/s = 162 ft/s

 

To get the speed in mi/hr: Δv = (162 ft/s)·(1 mi)/(5280 ft)·(3600 s)/(1 hr) = 110.5 mi/hr ≅ 110 mi/hr

 

 

 

Answer 2

     Kind of counter intuitive but the outer edge og something rotating is moving faster than the inner edge...cd's, tires....

Here's why. The tip of the blade is 119 ft from the center (hub has a radius of 3). It therefore travels around a circuference of c=pi*d =238pi feet in 4.5 seconds. the base of the blade is 3 units from the center and travels around a circuference of c=6*pi feet in 4.5 sec. So outer blade is moving 232 pi feet per4.5 seconds faster. Now convert to miles/hour.

If I did it right, approx. 110.4 mph.