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?

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Answer 1 · 2015-07-04T17:18:26

30° $rarr d=4.1/6pi$ m $~~2.1$ m

30rad $rarr d=123$ m

30rev $rarr d=246pi$ m $~~772.8$ m

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

$P=2pir=2pi*4.1=8.2pi$ 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
$30/360=3/36=1/12$ of this circle's perimeter, that is:

$1/12*8.2pi=8.2/12pi=4.1/6pi$ 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 $2pi$ rad, then an 30rad angle represents
$30/(2pi)=15/pi$ of this circle's perimeter, that is:

$15/pi*8.2pi = 15*8.2=123$ 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:

$30*8.2pi=246pi$ m