Angular Speed

Our world turns on wheels. Imagine rotating a wheel with its center at the origin, the rate at which the wheel is turning is its angular speed. The angular speed is often measured in revolutions per unit of time (for example, tachometers measure the speed of a car motor in revolutions per minute and bench grinders, circular saws and other high speed tools are often rated in revolutions per minute). In mathematics we prefer to use that powerful unit of measure, the radian, which we denote here by the symbol q . So we measure angular speed, w , in radians per unit of time, . A concept related to the angular speed of a spinning wheel is the linear velocity or speed of a point on the rim of the wheel (the actual distance the point travels per unit of time). The relationship is simple. A point on the rim of a wheel of radius r, spinning with an angular velocity of w radians per unit of time, has a linear velocity or speed of v length units per unit of time: . Another way of writing this formula is or simply

Let's try some problems from section 5.1 of the text.

1. A saw blade with diameter 7.5 inches is turning at 2400 revolutions per minute. What is the speed of the tip of the saw blade? In other words, how fast would the tip of the blade be moving if it left the blade?
Solution. An angular speed of 2400 revolutions per minute corresponds to 2400(2p ) = 4800p radians per minute. This means the linear velocity or speed is v = rw = 3.75(4800p ) = 56548.67 inches per minute. Ans. x 1 ft./12 in. x 1 min./60 sec. = 78.5398 feet per second.
Alternate Solution. Often we are better off converting the units to those desired in the end result. We have a radius, r, of 3.75 inches or 0.3125 feet. 2400 revolutions per minute corresponds to 2400/60 or 40 revolutions per second. Using these values, the angular speed, w , is 40(2p ) = 80p radians per second. The tip speed is v = rw = 0.3125(80p ) = 25p feet per second or approximately 78.5398 feet per second.
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3. The radius of the magnetic disk in a 3.5-inch diskette is 1.68 inches. Find the linear speed of a point on the circumference of the disk if it is rotating at a speed of 360 revolutions per minute.
Solution. Let v represent the speed of a point on the circumference of the disk.
The arc length of one revolution is s = 2p r = 2p (1.68) = 3.36p inches. The time in seconds, t, for one revolution is 1/6 sec. since 360 revolutions per minute is 6 revolutions per second. So v = 3.36p / (1/6) = 3.36p (6) = 20.16p in./sec. or approximately 63.33 in./sec.