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Chapter 5: Problem 66

A high-speed dental drill is rotating at $3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}$ Through how many degrees does the drill rotate in 1.00 s?

Short Answer

Expert verified
Answer: The dental drill rotates approximately \(1.80 \times 10^{6}\) degrees in 1.00 second.

Step by step solution

01

Convert radians per second to degrees per second

To convert radians to degrees, we will use the following conversion factor: \(1 \mathrm{rad} = \frac{180}{\pi} \mathrm{degrees}\). Given the angular speed in radians per second, \(3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}\), we will multiply it by the conversion factor. $$ \text{Angular speed in degrees per second} = (3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}) \times \frac{180}{\pi} \mathrm{degrees} $$
02

Calculate the total degrees rotated in 1.00 second

To find the total degrees rotated in 1.00 second, multiply the calculated angular speed in degrees per second by the time (1.00 s): $$ \text{Total degrees rotated} = (\text{Angular speed in degrees per second}) \times (1.00 \mathrm{s}) $$
03

Evaluate the expression

Calculate the angular speed in degrees per second and the total degrees rotated: $$ \text{Angular speed in degrees per second} = (3.14 \times 10^{4} \mathrm{rad}/\mathrm{s}) \times \frac{180}{\pi} \mathrm{degrees} \approx 1.80 \times 10^{6} \mathrm{degrees}/\mathrm{s} $$ $$ \text{Total degrees rotated} = (1.80 \times 10^{6} \mathrm{degrees}/\mathrm{s}) \times (1.00 \mathrm{s}) \approx 1.80 \times 10^{6} \mathrm{degrees} $$ So, the high-speed dental drill rotates approximately \(1.80 \times 10^{6}\) degrees in 1.00 second.

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