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Velocities Trigonometry MATH 103 S. Rook

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Overview Section 3.5 in the textbook: – Linear velocity – Angular velocity – The relationship between linear and angular velocity 2

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Linear Velocity

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Velocity: the rate at which an object travels – Expressed as a ratio of units of distance to units of time E.g. miles per hour, feet per second Linear Velocity (v): Given point P on a circle of radius r, the distance s traveled by P on the circumference of the circle in time t 4

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Linear Velocity (Example) Ex 1: Solve given that point P travels along the circumference of a circle: a) Find the linear velocity of point P when s = 100 miles and t = 4 hours b)Find the distance traveled by point P when v = 55 miles per hour and t = ½ hour 5

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Angular Velocity

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An alternative way to measure velocity is to use the central angle theta of a circle – In this interpretation, the velocity measures the number of radians covered in time t Angular Velocity (ω): Given point P on a circle with radius r, the central angle θ swept out in time t as point P moves around the circumference of the circle 7

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Angular Velocity (Continued) It is VERY important to understand whether a given velocity is either linear or angular: – Linear velocity is expressed as units of distance per unit of time – Angular velocity is expressed as radians per unit of time Revolution is a hidden format of angular velocity: – Recall that ω must have the units radians per unit of time – Revolution means one complete trip around a circle – Use the conversion ratio 8

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Angular Velocity (Example) Ex 2: Solve given that point P travels along the circumference of a circle: a) Find the angular velocity when θ = 24π and t = 1.8 hours b)Find the distance traveled when, r = 8 m, and t = 20 seconds 9

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Angular Velocity (Example) Ex 3: A lawnmower has a blade that extends out 2 feet from its center. The tip of the blade is traveling a 950 feet per second. Through how many revolutions per minute is the blade turning? 10

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The Relationship Between Linear and Angular Velocity

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Sometimes knowing how linear and angular velocities are related can make a problem easier to solve Consider the arc length formula: – If we divide both sides by t: Therefore, the relationship between linear velocity v and angular velocity ω is 12

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The Relationship Between Linear and Angular Velocity (Example) Ex 4: A cable railway is driven by a 10-foot diameter drum that turned at a rate of 20 revolutions per minute. Find the speed of the cable car, in miles per hour, by determining the linear velocity of the cable 13

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The Relationship Between Linear and Angular Velocity (Example) Ex 5: An engineering firm is designing a ski lift. The wire rope needs to travel with a linear velocity of 2.0 meters per second and the bullwheel should make 10 revolutions per minute. a) What is the angular velocity of the bullwheel? b) What diameter bullwheel should be used to drive the wire rope? 14

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Summary After studying these slides, you should be able to: – Solve problems involving linear velocity – Solve problems involving angular velocity – Solve problems involving both linear and angular velocity Additional Practice – See the list of suggested problems for 3.5 Next lesson – Basic Graphs (Section 4.1) 15

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