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Chapter 5: Problem 8

A wheel is rotating at a rate of 2.0 revolutions every \(3.0 \mathrm{s} .\) Through what angle, in radians, does the wheel rotate in \(1.0 \mathrm{s} ?\)

Short Answer

Expert verified
Answer: The wheel rotates \(\frac{4}{3}\pi\) radians in 1.0 seconds.

Step by step solution

01

Calculate the rate per second

First, we need to find out how many revolutions the wheel makes per second. Divide the number of revolutions by the given time to obtain the revolutions per second: Rate (revolutions per second) \(= \frac{2.0\,\text{revolutions}}{3.0\,\text{seconds}}\).
02

Simplify the rate

Now, simplify the fraction to get the rate as a simplified number: Rate (revolutions per second) \(= \frac{2}{3}\,\text{revolutions/s}\).
03

Convert revolutions to radians

In one full revolution, a wheel makes an angle of \(2 \pi\) radians. To convert the rate in revolutions/s to radians/s, simply multiply the rate by \(2\pi\): Rate (radians per second) \(= \frac{2}{3} \times 2\pi\,\text{radians/s}\).
04

Calculate the angle in 1 second

Now that we have the rate in radians per second, we can find the angle the wheel rotates in 1.0 seconds by multiplying the rate in radians/s by 1.0 seconds: Angle rotated in 1 second \(= \frac{2}{3}\times 2\pi\,\text{radians/s} \times 1.0\,\text{s}\).
05

Simplify the result

Perform the multiplication, simplifying the expression to obtain the final answer: Angle rotated in 1 second \(= \frac{4}{3}\pi\,\text{radians}\).

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