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Chapter 4: Problem 76

\(A\) fan motor turns at a given angular speed. How does the speed of the tips of the blades change when a fan of greater diameter is on the motor? Explain.

Short Answer

Expert verified
The speed of the tips of the blades increases when a fan of greater diameter is used on the motor. This increase is due to the direct proportionality between the radius (r) and the linear speed (v) in the equation \(v = \omega r\), where \(\omega\) is the constant angular speed of the motor.

Step by step solution

01

Understanding the Concept

Angular speed (\(\omega\)) is a measure of the rate of change in angle for a rotating object with respect to time. It remains constant for a fan motor. The equation \(v = \omega r\) shows that the linear speed at a point on the edge of the rotating body (like the tip of a fan blade) depends on both the angular speed and the distance of the point from the center of rotation (the radius \(r\)).
02

Using Given Information

The exercise states that the fan motor turns at a given angular speed which means the value of \(\omega\) is constant. When a larger diameter fan is used, the radius of the fan (\(r\)) increases even though the angular speed remains the same.
03

Calculating New Blade Tip Speed

As \(v = \omega r\), when \(r\) increases (i.e., a fan with a larger diameter is used), the linear speed experienced by the tips of the blades (\(v\)) must increase to maintain the equation, assuming \(\omega\) remains constant. Therefore, the speed of the tips of the blades increases when a fan of greater diameter is used on the motor.

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