Pre-Calc Exam Notes 101

Pre-Calc Exam Notes 101 - central angle does it sweep out...

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Circular Motion: Linear and Angular Speed Section 4.4 101 Example 4.14 An object sweeps out a central angle of π 3 radians in 0.5 seconds as it moves along a circle of radius 3 m. Find its linear and angular speed over that time period. Solution: Here we have t = 0.5 sec, r = 3 m, and θ = π 3 rad. So the angular speed ω is ω = θ t = π 3 rad 0.5 sec ω = 2 π 3 rad/sec , and thus the linear speed ν is ν = ω r = p 2 π 3 rad/sec P (3 m) ν = 2 π m/sec . Note that the units for ω are rad/sec and the units of ν are m/sec. Recall that radians are actually unitless, which is why in the formula ν = ω r the radian units disappear. Example 4.15 An object travels a distance of 35 ft in 2.7 seconds as it moves along a circle of radius 2 ft. Find its linear and angular speed over that time period. Solution: Here we have t = 2.7 sec, r = 2 ft, and s = 35 ft. So the linear speed ν is ν = s t = 35 feet 2.7 sec ν = 12.96 ft/sec , and thus the angular speed ω is given by ν = ω r 12.96 ft/sec = ω (2 ft) ω = 6.48 rad/sec . Example 4.16 An object moves at a constant linear speed of 10 m/sec around a circle of radius 4 m. How large of a
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Unformatted text preview: central angle does it sweep out in 3.1 seconds? Solution: Here we have t = 3.1 sec, = 10 m/sec, and r = 4 m. Thus, the angle is given by s = r = s r = t r = (10 m/sec)(3.1 sec) 4 m = 7.75 rad . In many physical applications angular speed is given in revolutions per minute , abbrevi-ated as rpm . To convert from rpm to, say, radians per second, notice that since there are 2 radians in one revolution and 60 seconds in one minute, we can convert N rpm to radians per second by canceling the units as follows: N rpm = N rev min 2 rad 1 rev 1 min 60 sec = N 2 60 rad/sec This works because all we did was multiply by 1 twice. Converting to other units for angular speed works in a similar way. Going in the opposite direction, say, from rad/sec to rpm, gives: N rad/sec = N 60 2 rpm...
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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