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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]

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.