math culminating

**VDestinV** · June 16th 2008, 12:21 PM

A carousel at a local fall fair makes three roations in 1 min. The caraousel is 10 m in diameter. A ride lasts 5 minutes.

a) Determine the angular velocity in rad/min and in rads.

b) How far does a child who is riding the carousel travel in 5 minutes?

**topsquark** · June 16th 2008, 03:30 PM

Originally Posted by VDestinV

A carousel at a local fall fair makes three roations in 1 min. The caraousel is 10 m in diameter. A ride lasts 5 minutes.

a) Determine the angular velocity in rad/min and in rads.

b) How far does a child who is riding the carousel travel in 5 minutes?

I presume the last part in a) is rad/s?

a) Angular speed is measured as "angle per time." We know it makes three revolutions (rotations) in one minute. The rest is unit conversion.
$\frac{3~revolutions}{1~min} \cdot \frac{2 \pi~rad}{1~revolution} = 6 \pi~rad/min$

Then another unit conversion:
$\frac{6 \pi~rad}{1~min} \cdot \frac{1~min}{60~s} = \frac{6 \pi~rad}{60~s} = \frac{\pi}{10}~rad/s$

Since the time is in minutes, let's use the angular speed in rad/min. We know that the angular speed is constant, so the angle swept out by the child is
$\omega = \frac{\Delta \theta}{\Delta t}$

$\Delta \theta = \omega \Delta t = (6 \pi~rad/min)(5~min) = 30 \pi~rad$

Then the distance traveled will be
$\Delta s = r \Delta \theta = (5~m)(30 \pi~rad) = 150 \pi~m$

-Dan

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