Applications of Radian Measure

Recall Radian measure: One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. Algebraically, this means that where θ is measured in radians.

Example 1 Find the radian measure of the central angle of a circle of radius 27 inches that intercepts an arc of length 6 inches.

Example 2 Find the length of the arc on a circle of radius 9 feet intercepted by a central angle 60°.

Example 3 Find the distance between the two cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other). Dallas, TX 32°47ʹ39ʺN Omaha, NE 41°15ʹ50ʺN

Example 4 Find the area of the sector of the circle with radius 4 inches and central angle 225°.

Example 5 Find the area of the sector of the circle with radius 12 m and central angle π/4.

Linear & Angular Velocity Definition: Linear Velocity is distance/time: Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec Definition: Angular Velocity is central angle/time: Ex. 360°/day, 2π rad/hour 30°/min

Linear & Angular Velocity Deriving a formula which relates the two velocities: Definition of Linear Velocity: Recall Arc Length Formula Recognize Definition of Angular Velocity:

Example 6 Consider the Earth which rotates on its axis once every 24 hours. Find the angular velocity of the Earth’s rotation in a) radians per hour and b) degrees per hour.

Example 7 Find the linear velocity of a point on the surface of the Earth. Assume that the radius of the Earth is 4000 miles.

Example 8 Find the angular velocity of a ceiling fan if the fan rotates 40 times per minute.

Example 9 Calculate the linear velocity of the fan blade if the radius of the fan is 2 feet.

Example 10 A carousel with a 50 ft diameter makes 4 revolutions per minute. a) Find the angular speed of the carousel in radians per minute b) Find the linear speed of the platform rim of the carousel.