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?

1 Answer
Jul 4, 2015

30° rarr d=4.1/6pi m ~~2.1m

30rad rarr d=123m

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

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 2pirad, then an 30rad angle represents
30/(2pi)=15/pi of this circle's perimeter, that is:

15/pi*8.2pi = 15*8.2=123m

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