## Chapter 7

## Q. 7.1

## Q. 7.1

**WHIRLYBIRDS**

**GOAL** Perform some elementary calculations with angular variables.

**PROBLEM** The rotor on a helicopter turns at an angular velocity of 3.20 × 10² revolutions per minute. (In this book, we sometimes use the abbreviation rpm, but in most cases we use rev/min.) (**a**) Express this angular velocity in radians per second. (**b**) If the rotor has a radius of 2.00 m, what arclength does the tip of the blade trace out in 3.00 × 10² s? (**c**) The pilot opens the throttle, and the angular velocity of the blade increases while rotating twenty-six times in 3.60 s. Calculate the average angular velocity during that time.

**STRATEGY** During one revolution, the rotor turns through an angle of 2π radians. Use this relationship as a conversion factor. For part (**b**), first calculate the angular displacement in radians by multiplying the angular velocity by time. Part (**c**) is a simple application of Equation 7.3.

\omega_{av} \equiv \frac{\theta_f-\theta_i}{t_f-t_i} =\frac{\Delta\theta}{\Delta t} [7.3]

## Step-by-Step

## Verified Solution

(**a**) Express this angular velocity in radians per second.

Apply the conversion factors 1 \mathrm{rev}=2 \pi rad and 60.0 \mathrm{~s}=1 \mathrm{~min}:

\begin{aligned}\omega &=3.20 \times 10^2 \frac{\mathrm{rev}}{\mathrm{min}} \\&=3.20 \times 10^2 \frac{\cancel{\mathrm{rev}}}{\cancel{\mathrm{min}}}\left( \frac{2\pi \mathrm{~rad}}{1 \cancel{\mathrm{~rev}}} \right)\frac{1.00 \cancel{\mathrm{~min}}}{60.0 \mathrm{~s}} \\&=33.5 \mathrm{~rad} / \mathrm{s}\end{aligned}

(**b**) Find the arclength traced out by the tip of the blade.

Multiply the angular velocity by the time to obtain the angular displacement:

\Delta \theta=\omega t=(33.5 \mathrm{~rad} / \mathrm{s})\left(3.00 \times 10^2 \mathrm{~s}\right)=1.01 \times 10^4 \mathrm{~rad}

Multiply the angular displacement by the radius to get the arclength:

\Delta s=r \Delta \theta=(2.00 \mathrm{~m})\left(1.01 \times 10^4 \mathrm{rad}\right)=2.02 \times 10^4 \mathrm{~m}

(**c**) Calculate the average angular velocity of the blade while its angular velocity increases.

Apply Equation 7.3, noticing that

\begin{aligned}&\Delta \theta=(26 \mathrm{rev})(2 \pi \mathrm{rad} / \mathrm{rev})=52 \pi \mathrm{rad} \\&\omega_{\mathrm{av}}=\frac{\Delta \theta}{\Delta t}=\frac{52 \pi \mathrm{rad}}{3.60 \mathrm{~s}}=45 \mathrm{rad} / \mathrm{s}\end{aligned}

**REMARKS** It’s best to express angular velocity in radians per second. Consistent use of radian measure minimizes errors.