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