SECTION 6.1 Angles, Arc Length, and Circular Motion 391 Angular speed is the way the turning rate of an engine is described. For example, an engine idling at 900 rpm (revolutions per minute) is one that rotates at an angular speed of π π = ⋅ = 900 revolutions minute 900 revolutions minute 2 radians revolution 1800 radians minute There is an important relationship between linear speed and angular speed: v s t r t r t r linear speed θ θ ω = = = = ⋅ = ⋅ ↑ ↑ ↑ (9) s rθ = (10) DEFINITION Angular Speed The angular speed ω (the Greek lowercase letter omega) of an object is the angle θ (measured in radians) swept out, divided by the elapsed time t ; that is, t ω θ = (10) v rω = (11) where ω is measured in radians per unit time. When using formula (11), v r ,ω = remember that v s t = (the linear speed) has the dimensions of length per unit of time (such as feet per second or miles per hour), r (the radius of the circular motion) has the same length dimension as s, and ω (the angular speed) has the dimension of radians per unit of time. If the angular speed is given in terms of revolutions per unit of time (as is often the case), be sure to convert it to radians per unit of time, using the fact that 1revolution 2π = radians, before using formula (11). Finding Linear Speed A person is spinning a rock at the end of a 2-foot rope at the rate of 180 revolutions per minute (rpm). Find the linear speed of the rock when it is released. Solution Look at Figure 19. The rock is moving around a circle of radius r 2feet. = The angular speed ω of the rock is ω π π = = ⋅ = 180 revolutions minute 180 revolutions minute 2 radians revolution 360 radians minute From formula (11), v r ,ω = the linear speed v of the rock is ω π π = = ⋅ = ≈ v r 2 feet 360 radians minute 720 feet minute 2262 feet minute The linear speed of the rock when it is released is 2262ft min 25.7mi hr. ≈ Now Work PROBLEM 99 EXAMPLE 8 Figure 19 r 5 2
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