An electric ceiling fan is rotating about a fixed axis with an initial angular v

Question

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.210({ m rev/s}) . The magnitude of the angular acceleration is 0.909({ m rev/s^2}) . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.790({ m m}) .

Part D

What is the magnitude ( exttip{a}{a}) of the resultant acceleration of a point on the tip of the blade at time ( exttip{t}{t}) = 0.201({ m s}) ?

Express the acceleration numerically in meters per second squared.

Explanation / Answer

D)

radial  acceleration = r*w^2

where r = radius = diameter/2 = 0.79/2 = 0.395 m

At time t = 0.201

we have angular velocity w = w0 + t* angular acceleration

w = 0.21 + (0.909 * 0.201) = 0.392709 rev/s = 0.392709 * 2* pi rad/s = 2.46746 rad/s

so  radial acceleration = r*w^2 = 0.395 * 2.46746^2 = 2.4049 m/s^2 is the answer

we have tangentaial accelration = .795 m/s^2

so resulatnt acceleration = sqrt ( 0.795^2 + 2.4049^2) = 2.5328 m/s^2 is the answer

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