Constants Modern wind turbines are larger than they appear, and despite their apparently lazy motion, the speed of the blades tips can be quite high-many times higher than the wind speed. A turbine has blades 58 m long that spin at 14 rpm . At the tip of a blade, what is the centripetal acceleration?

During the motion, the turbine will undergo rotational motion. Hence the radius of the circle traced by the turbine is equal to the length of the blade.

Hence radius r = 58 m.

The frequency of the turbine [f] =14 rpm.

Here rpm stands for rotation per minute.

Hence the frequency of the turbine in one second-

[tex]f=\frac{14}{60}\ s^-1[/tex]

[tex]f=0.23333 Hz[/tex]

Here Hz[ hertz] is the unit of frequency.

The angular velocity of the turbine [tex]\omega =2\pi f[/tex]

[tex]\omega=2*3.14*0.2333[/tex]

[tex]\omega=1.465124[/tex] radian/second

Now we have to calculate the centripetal acceleration of the blade.

Let the linear velocity of the blade is v.

we know that linear velocity v=ωr

The centripetal acceleration is calculated as-

[tex]a_{c} =\frac{v^2}{r}[/tex]

[tex]=\frac{[\omega r]^2}{r}[/tex]

[tex]=\omega^2r[/tex]

[tex]=[1.465124]^2 *58[/tex]

[tex]=124.5021234 m/s^2[/tex] [ans]

snehashish65 snehashish65 · Answer 2 · 2022-03-22T11:22:55+01:00

Centripetal acceleration depends on the rotational speed of object. The required value of the centripetal acceleration of the wind turbine blade is [tex]125.33 \;\rm rad/s^{2}[/tex].

What is centripetal acceleration?

The magnitude of acceleration possessed by any object with rotational motion, such that the direction of acceleration is towards the center, then such acceleration is called centripetal acceleration.

Given data-

The length of the turbine blade is, r = 58 m.

The angular speed of blade is, N = 14 rpm.

Convert the angular speed into rad/s as,

[tex]\omega = \dfrac{2 \pi N}{60}\\\\\\\omega = \dfrac{2 \pi \times 14}{60}\\\\\omega =1.47 rad/s[/tex]

And the expression for the angular acceleration is given as,

[tex]\alpha = \omega^{2} \times r[/tex]

Solving as,

[tex]\alpha = 1.47^{2} \times 58\\\\\alpha =125.33 \;\rm rad/s^{2}[/tex]

Thus, we can conclude that the required value of the centripetal acceleration of wind turbine blade is [tex]125.33 \;\rm rad/s^{2}[/tex].

Learn more about the centripetal acceleration here:

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