# A wheel has a radius of 4.1m. How far(path length) does a point on the circumference travel if the wheel is rotated through angles of 30° , 30 rad, and 30 rev, respectively?

Jul 4, 2015

30° $\rightarrow d = \frac{4.1}{6} \pi$ m $\approx 2.1$m

30rad $\rightarrow d = 123$m

30rev $\rightarrow d = 246 \pi$ m $\approx 772.8$m

#### Explanation:

If the wheel has a 4.1m radius, then we can calculate its perimeter:

$P = 2 \pi r = 2 \pi \cdot 4.1 = 8.2 \pi$ m

When the circle is rotated through an 30° angle, a point of its circumference travels a distance equal to an 30° arc of this circle.

Since a full revolution is 360°, then an 30° arc represents
$\frac{30}{360} = \frac{3}{36} = \frac{1}{12}$ of this circle's perimeter, that is:

$\frac{1}{12} \cdot 8.2 \pi = \frac{8.2}{12} \pi = \frac{4.1}{6} \pi$ m

When the circle is rotated through an 30rad angle, a point of its circumference travels a distance equal to an 30rad arc of this circle.

Since a full revolution is $2 \pi$rad, then an 30rad angle represents
$\frac{30}{2 \pi} = \frac{15}{\pi}$ of this circle's perimeter, that is:

$\frac{15}{\pi} \cdot 8.2 \pi = 15 \cdot 8.2 = 123$m

When the circle is rotated through an 30rev angle, a point of its circumference travels a distance equal to 30 times its perimeter, that is:

$30 \cdot 8.2 \pi = 246 \pi$ m