SOLUTION: A wheel had an angular velocity of 400 radians per minute. How fast was the wheel rolling along in kilometers per hour if the radius of the wheel was 160 centimeters?

Question 302547: A wheel had an angular velocity of 400 radians per minute. How fast was the wheel rolling along in kilometers per hour if the radius of the wheel was 160 centimeters?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
If the number of degrees per minute is in radians, then the formula should be:

velocity = angular velocity * radius

If that's true, then:

angular velocity = 400 radians per minute.
radius = 160 centimeters.

velocity = 400 * 160 = 64000 centimeters per minute.

Since there are 60 minutes in an hour, then this is equivalent to 3,840,000 centimeters per hour.

Since there are 100,000 centimeters in a kilometer, then this is equivalent to 3.84 kilometers per hour.

This formula assumes that the angular velocity is expressed in radians per unit of time (minutes in this case).

If you did not know the formula, then you would have had to rationalize the problem as follows:

In general, the linear velocity is equal to the number of rotations of the wheel per minute times the circumference of the wheel.

The circumference of the wheel is equal to 2*pi*r = 2*pi*160 = 1005.309649 centimeters.

Since 1 complete rotation of the wheel is equal to 2*pi radians (360 degrees), then the number of rotations of the wheel per minute is equal to the number of radians the wheel is spinning per minute divided by 2*pi.

400/2*pi = 63.66197724 rotations of the wheel per minute.

63.66197724 * 1005.309649 = 64000 centimeters of linear travel per minute.

Since there are 60 minutes in an hour, then this is equivalent to 3,840,000 centimeters per hour.

Since there are 100,000 centimeters in a kilometer, then this is equivalent to 3.84 kilometers per hour.

You could also have converted everything to degrees and gotten the same answer as follows:

Since number of degrees is equal to number of radians * 180 / pi, then 400 radians per minute of rotation is equivalent to 22918.31181 degrees per minute of rotation.

Number of rotations of the wheel per minute is equal to number of degrees of rotation per minute / 360 = 63.66197724 rotations of the wheel per minute.

That gets you the same answer.

You just had to know that the linear velocity of the wheel is equal to the number of rotations of the wheel per minute times the circumference of the wheel.

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