Question

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

In terms of tangential velocity, the tip is traveling 110.432 miles per hour faster than the root of the windmill blade.

Explanation:

The first step is to determine the angular velocity of the windmill's rotor.

Angular Velocity is given in radians per second. Since the windmill makes a full rotation in 4.5 seconds, and one rotation is `2pi` radians, then the angular velocity (`Omega`) is:

`Omega=(2pi)/4.5 (rad)/s=color(blue)((4pi)/9 (rad)/s`

To find the Tangential Velocity, or the linear velocity at a given radius, we need simply to multiply the Angular Velocity by the desired radius:

`V_t=Omegaxxr`

To determine the difference in Tangential Velocities, we would simply subtract one from the other. Since both points are traveling at the same Angular velocity, we can simplify the equation somewhat:

`Delta V_t = V_"Tip"-V_"Base"`

`Delta V_t = color(blue)(Omega)xxr_"Tip"-color(blue)(Omega)xxr_"Base"`

`Delta V_t = color(blue)(Omega)xx(r_"Tip"-r_"Base")`

Now, because we've written the equation this way, knowing the discrete radii of the base and tip is irrelevant. We only need to know the distance between the radii.

Based on the problem statement, we know the length of the blade, which is the distance we need: 116 feet.

`r_"Tip"-r_"Base"=color(red)(116ft)`

`Delta V_t = color(blue)(Omega)xxcolor(red)(116)`

`Delta V_t = color(blue)((4pi)/9)xxcolor(red)(116)`

`Delta V_t = (464pi)/9 (ft)/s`

Finally, now that we have the delta-velocity in feet per second, we need to convert to miles per hour:

`cancel(ft)/cancel(s) xx (3600cancel(s))/(1hr) xx (1mi)/(5280cancel(ft))`

`rArr 3600/5280 = 15/22 (mi)/(hr)`

`Delta V_t = (464pi)/9 xx15/22`

`Delta V_t = (6960pi)/198 ~= 110.432 mph`