Question #b625e

Jan 1, 2017

$56.23$ mph, rounded to two decimal places.

Explanation:

For circular motion we have the following equation connecting linear velocity $v$, of a point located at radius $r$, and rotating with an angular frequency $\omega$

$v = r \omega$ ........(1)
and $\omega = 2 \pi f$ .....(2)
where $f$ is frequency of rotation.

Combining (1) and (2) we get
$v = r \left(2 \pi f\right)$
Inserting given values after converting inch to mile and minute to hour

1. For $14$​-inch diameter blade: $r = \frac{14}{2} = 7$ inch
${v}_{14} = \left(\frac{7}{12 \times 3 \times 1760}\right) \left(2 \pi \times \frac{3150}{\frac{1}{60}}\right)$
$\implies {v}_{14} = 7 \times \frac{2 \pi \times 3150 \times 60}{12 \times 3 \times 1760}$
$= 131.20$ mph
2. Similarly for $8$​-inch diameter blade: $r = \frac{8}{2} = 4$ inch
$\implies {v}_{8} = 4 \times \frac{2 \pi \times 3150 \times 60}{12 \times 3 \times 1760}$
$= 74.97$ mph

Difference in linear velocity of two$= {v}_{14} - {v}_{8}$
$= 131.20 - 74.97$
$= 56.23$ mph, rounded to two decimal places.