Question

A wind turbine is rotating counterclockwise at 0.5 rev/s and slows to a stop in 10 s. Its blades are 20 m in length. (a) What is the angular acceleration of the turbine? (b) What is the centripetal acceleration of the tip of the blades at t = 0 s? (c) What is the magnitude and direction of the total linear acceleration of the tip of the blades at t = 0 s?

Answer 1

Answer:

-0.314 rad/s²

197.39 m/s²

197.49 m/s² and 1.822° counterclockwise.

Explanation:

$\omega_f$ = Final angular velocity

$\omega_i$ = Initial angular velocity

$\alpha$ = Angular acceleration

$\theta$ = Angle of rotation

t = Time taken

R = Radius = 20 m

$\omega_i=0.5\times 2\pi\\\Rightarrow \omega_i=\pi$

$\omega_f=\omega_i+\alpha t\\\Rightarrow \alpha=\frac{\omega_f-\omega_i}{t}\\\Rightarrow \alpha=\frac{0-\pi}{10}\\\Rightarrow a=-0.314\ rad/s^2$

Angular acceleration of the turbine is -0.314 rad/s²

Centripetal acceleration

$a_c=\omega_i^2R\\\Rightarrow a_c=(\pi)^2\times 20\\\Rightarrow a_c=197.39\ m/s^2$

The centripetal acceleration of the tip is 197.39 m/s²

Tangential acceleration

$a_t=\alpha R\\\Rightarrow a_t=-0.314\times 20\\\Rightarrow a_t=-6.28\ m/s^2$

Acceleration

$a=\sqrt{a_c^2+a_t^2}\\\Rightarrow a=\sqrt{197.39^2+(-6.28)^2}\\\Rightarrow a=197.49\ m/s^2$

$\theta=tan^{-1}\frac{|a_t|}{|a_c|}\\\Rightarrow \theta=tan^{-1}\frac{6.28}{197.39}\\\Rightarrow \theta=1.822^{\circ}$

Magnitude and direction of the total linear acceleration of the tip of the blades is 197.49 m/s² and 1.822° counterclockwise.