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Angular acceleration is a measure of how quickly the angular speed of an object changes with time. Think of it as the rotational counterpart to linear acceleration. Just like a car accelerating in a straight line, a rotating object like a dentist's drill increases its angular speed over time. In this exercise, the drill accelerates from a stop to 350,000 revolutions per minute in just 2.1 seconds.

To find the angular acceleration, we first need to convert the final angular speed from rpm to radians per second, using the conversion factor where 1 revolution equals \(2\pi\) radians. This gives us the value \(\omega = 36651.85\) rad/s. Angular acceleration \(\alpha\) is then calculated using the formula:

• \(\alpha = \frac{\Delta \omega}{\Delta t}\)
• \(\Delta \omega = 36651.85\) rad/s – 0 rad/s = 36651.85 rad/s
• \(\Delta t = 2.1\) seconds

Plugging in these values, we calculate \(\alpha \approx 17453.26\) rad/s², indicating the drill's dramatic increase in speed per second.

Angular speed conversion is crucial for understanding rotational motion, particularly in physics and engineering applications. The standard unit of angular speed in these fields is radians per second. Sometimes, though, certain devices, like drills, express their speed in revolutions per minute (rpm).

To convert from rpm to radians per second, follow this formula:

• Multiply the given speed in rpm by \(\frac{2\pi}{60}\).
• For instance, 350,000 rpm becomes \(350,000 \times \frac{2\pi}{60} = 36651.85\) rad/s.

This conversion simplifies further calculations, like finding angular acceleration and displacement during rotational motion. Switching seamlessly between units allows for greater clarity and precision in solving problems related to angular kinematics.

Calculating the number of revolutions a rotating object makes involves understanding its angular displacement. Angular displacement tells us how far an object has rotated over a period of time in terms of radians.

For the drill coming up to speed in the problem, we use the angular kinematics formula to find the angular displacement \(\theta\) during acceleration:

• \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\)
• Given \(\omega_0 = 0\), \(\alpha = 17453.26\) rad/s², and \(t = 2.1\) seconds
• \(\theta = \frac{1}{2} \cdot 17453.26 \cdot (2.1)^2 \approx 38452.35\) rad

To convert radians to revolutions, remember that one revolution equals \(2\pi\) radians. Thus, the drill completes approximately \(\frac{38452.35}{2\pi} \approx 6120.9\) revolutions while accelerating to its operating speed.