Multiple-Concept Example 7 explores the approach taken in problems such as this

Question

Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.372 m and are rotating about a fixed axis with an angular velocity of +1.59 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of 1.88 rad/s2. After 0.510 s have elapsed since the switch was reset, what are the following? (a) the total acceleration (in m/s2) of a point on the tip of a blade m/s2 (b) the angle between the total acceleration and the centripetal acceleration ac °

Explanation / Answer

Angular acceleration =1.88rad/s²
After 0.510 s the angular velocity = 1.59 +1.88*0.510 = 2.55 rad/s

After 0.510 s the centripetal acceleration ac = r² = 0.372*2.55² = 2.42 m/s ²
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Linear acceleration aT = r = 0.372*1.88 = 0.7 m/ s²
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Total acceleration a (total) = [2.42² + 0.7²] = 2.52 m/s ²

If is the angle between the total acceleration and the centripetal acceleration

tan = aT / ac = 0.7 / 2.42

= 16.13°

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