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The spinning of a disk on a shaft often translates to linear motion. The most obvious example is an automobile wheel, but forward motion can also be important when designing gear and belt systems. The translation from rotation to linear speed is straightforward; all you need to know is the radius (or diameter) of the spinning disk. If you want the linear speed in feet per minute, it’s important to remember you need to measure the radius in feet.
TL;DR (Too Long; Didn’t Read)
For a disk spinning at _n_ rpm, the forward speed of the attached shaft is _n_ × 2π_r_ if the radius of the disk is _r_.
The Basic Calculation
Designate a point P on the circumference of a spinning disk. P makes contact with the surface once with each spin, and with each spin, it travels a distance equal to the circumference of the circle. If the frictional force is sufficient, the shaft attached to the disk moves forward that same distance with each rotation. A disk with radius r has a circumference of 2πr, so each rotation moves the shaft forward that distance. If the disk spins n times per minute, the shaft moves a distance n × 2πr each minute, which is its forward speed (s).
(s = n × 2πr)
It’s more common to measure the diameter (d) of a disk, such as a car wheel, than the radius. Since r = d ÷ 2, the forward speed of the car becomes nπd, where n is the rotational speed of the tire.
(s = n × πd)
Example
A car with 27-inch tires is traveling 60 miles per hour. How fast are its wheels spinning?
Convert the car’s speed from miles per hour to feet per minute: 60 mph = 1 mile per minute, which in turn is 5,280 ft/min. The car’s tire has a diameter of 1.125 feet. If s = n × πd, divide both sides of the equation by πd:
(n = frac{s}{πd} \,= frac{5280 text{ ft/min}}{3.14 × 1.125 text{ ft}} \,= 1,495 text{ rpm})
Friction Is a Factor
When a disk in contact with a surface rotates, the shaft around which the disk is spinning moves forward only if the force of friction between the disk and surface is large enough to prevent slipping. The frictional force depends on the coefficient of friction between the two surfaces in contact and on the downward force exerted by the weight of the disk and the weight applied to the shaft. These create a perpendicular downward force at the point of contact called the normal force, and this force becomes less when the surface is inclined. A car’s tires can start slipping when the car climbs a hill, and they can slip on ice, because the coefficient of friction of ice is less than that of asphalt.
Slippage affects forward motion. When translating rotational speed into linear speed, you can compensate for slippage by multiplying by an appropriate factor derived from the coefficient of friction and the incline angle.
