Question

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?

Answer 1

30° `rarr d=4.1/6pi` m `~~2.1`m

30rad `rarr d=123`m

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

Explanation:

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