In Example 4.15 part b, suppose a 52 cm diameter blade was attached to the motor shaft instead of the 76 cm diameter blade. If the fan blade experienced the same angular acceleration, what would happen to the magnitude of the tangential acceleration at the tip of the blade during the first few seconds?

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BlackTom AI · Accepted Answer

Question restatement: In Example 4.15 part b, if a 52 cm diameter blade is attached to the motor shaft instead of the 76 cm blade, and the fan blade experiences the same angular acceleration, what happens to the magnitude of the tangential acceleration at the tip of the blade during the first few seconds?
Option 1: It would be larger with the 52 cm blade. This would be incorrect because tangential acceleration a_t = r * alpha, where r is the radius and alpha is the angular acceleration. Since the 52 cm blade has a smaller radius than 76 cm, keeping alpha the same would yield a smaller tangential acceleration, not larger.
Option 2: It would be smaller with the 52 cm blade. This is the correct line of reasoning: a_t = r * alpha, so reducing r (the blade tip distance from the axis) directly reduces the tangential acceleration when alpha remains unchanged. Therefore, the smaller blade radius results in a smaller a_t.
Option 3: It would not change. This is incorrect because tangential acceleration depends on the radius: a_t scales linearly with r for a given angular acceleration. Halving or reducing the radius will reduce a_t accordingly, so it cannot stay the same when alpha is fixed.
Additional clarification: The key relationship here is a_t = r * alpha. When the only change is blade diameter (hence radius), and the angular acceleration is fixed, the tangential acceleration at the tip scales with the radius. A smaller radius directly reduces a_t, which aligns with option 2.

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