Chapter 6.1 – angular and linear velocity 6.1 Angle Measures Angular and Linear Velocity and Arc Length a few simple/silly word problems… Yao Ming is about 8 feet tall. The local water tower has a height of about 5 “Yao Ming’s”. How many feet tall is the water tower? a few simple/silly word problems… My friend’s “soccer mom minivan” has a length of 14 feet (it’s HUGE). My friend’s house has a length of 10 “soccer mom minivans”. How long is my friend’s house? (yes this is silly but there’s a point to this) so this is where it gets real (and why we use radians) The earth has a radius of about 4000 miles. An airplane flies from North America to Europe, and covers an angular distance of 2.0 radians. (that’s a little less than 120 degrees) How many miles did the plane travel? (hint: 2.0 radians means 2.0 “radiuses”!) and one more… (this one requires some dimensional analysis) A wagon wheel rolls down a hill and, in the process, makes 10 full revolutions (each revolution is 2π radians). If the wheel has a radius of 30 inches, how many FEET did the wheel roll down the hill? Chapter 6.1 – angular and linear velocity Length of a Circular Arc s r Θ has to be in radians!! • Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30o. • A central angle θ in a circle of radius 4 m is subtended by an arc of length 6m. Find the measure of θ in radians. Chapter 6.1 – angular and linear velocity Area of a Circular Sector 1 2 A r 2 Θ has to be in radians!! • Find the area of a sector of a circle with central angle 60o if the radius of the circle is 3 m. Chapter 6.1 – angular and linear velocity Circular Motion • Angular Speed (ω): Always a circular measurement over time. – degrees per second – Radians per minute – Revolutions per minute (RPM) – Rotations per hour Chapter 6.1 – angular and linear velocity Circular Motion • Linear Speed (v): Always a linear measurement over time. – – – – • mph (miles per hour) Feet per second Inches per minute Kilometers per hour v r r=radius ω=angular velocity Chapter 6.1 – angular and linear velocity Jon and Kate are riding on a Ferris wheel. Jon doesn’t want to ride with Kate, and gets off of the ride. He observes that it takes 50 seconds to make a complete revolution (that is, to travel one complete circle). Their seat is 30 feet from the axle of the wheel. a) What is their angular velocity in revolutions per minute? b) in degrees per minute? c) in radians per minute? Chapter 6.1 – angular and linear velocity Kate gets mad at Jon, and puts a rock in her sling and starts whirling it around. She realizes that in order for the rock to hit Jon, it must leave her sling at a speed of 100 feet per second. (is this a measure of linear velocity or angular velocity?) Her sling is swung in a circular path of radius of 5 feet. What must the angular velocity be in order for Kate’s rock to reach Jon? Note: the McNeil math department strong discourages the use of slingshots and stones in the settling of domestic disputes. Chapter 6.1 – angular and linear velocity Jon gets in his car to get the heck away from Kate. The rear wheels on his 350z sports car spin at a rate of 5000 revolutions per minute. If a wheel has a radius of 9 inches, how fast is the wheel turning in miles per hour? (this one involves a lot of conversions. Hint: 1 mile = 5280 feet) Please buckle up and drive safely. Getting hit by a rock hurts… but so does getting into a car accident.