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 0.260 rev/s . The magnitude of the angular acceleration is 0.907 rev/s2 . Both the the angular velocity and angular accleration are directed clockwise. The electric ceiling fan blades form a circle of diameter 0.790 m .

A) Compute the fan's angular velocity magnitude after time 0.191 s has passed.

B) Through how many revolutions has the blade turned in the time interval 0.191 s from Part A?

C) What is the tangential speed vtan(t) of a point on the tip of the blade at time t = 0.191 s ?

Express your answer numerically in meters per second.

D) What is the magnitude a of the resultant acceleration of a point on the tip of the blade at time t = 0.191 s ?

Express the acceleration numerically in meters per second squared.

Explanation / Answer

A)

What does an acceleration of 0.907 rev/s² *mean*?

It means that after one second, the ceiling fan will be going 0.907 revolutions per second faster, and after two seconds, it will be going 1.814 revolutions per second faster, and so on.

In other words, after some time t, the angular velocity will be:

v2 = v1 + (0.907 rev/s²) * t.

Where v1 is the initial velocity (in this case, 0.260 rev/s).

v2 = 0.260 + (0.907*0.191) = 0.433237 rev/s

For problem (B)

, you need to know that the number of revolutions N is given by:

N = t * (v1 + v2)/2 = 0.191*(0.260 + 0.433237)/2 = 0.066204133 turns.

You can only do this with a *constant* angular acceleration. The idea is to take the *average* of the initial and final angular velocities, and that's the "average speed", as it were. So the "distance" is the "average speed" times the time interval.

For (C),

remember that one revolution sweeps out a distance of 2 * pi * r, where r is apparently 0.790/2 meters. That means that you can take your answer for (1) and go:

(revolutions per second) * (meters per revolution) = (meters per second).

For (D),

you need to just know that the *inward* acceleration of something that's moving in a circular motion is a_i = v² / r.

But that's not the only acceleration: there's also a tangential acceleration, a_t.

But fortunately, you can just use 0.907 rev/s² for that, once you convert "revolutions" to "meters", the same way that you did in part (3).

These two accelerations are perpendicular. The magnitude of the total acceleration is given by the Pythagorean theorem as:

a = sqrt( (a_i)² + (a_t)² ).

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