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
Olivier B.
Jul 4, 2015

30° →d=4.16π m ≈2.1m

30rad →d=123m

30rev →d=246π m ≈772.8m

Explanation:

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

P=2πr=2π⋅4.1=8.2π 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
30360=336=112 of this circle's perimeter, that is:

112⋅8.2π=8.212π=4.16π 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 2πrad, then an 30rad angle represents
302π=15π of this circle's perimeter, that is:

15π⋅8.2π=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⋅8.2π=246π m

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