Important variables for problems in which an object is moving along a circular arc

Size: px

Start display at page:

Download "Important variables for problems in which an object is moving along a circular arc"

Norma Shields
6 years ago
Views:

1 Unit - Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side - starting and ending Position of the ray Standard position origin if the vertex, initial side is the positive x-axis Angle: rotating a ray about its endpoint Vertex endpoint of the ray Positive, negative angles positive is counterclockwise rotation, negative axis clockwise rotation Co-terminal angles angles having the same initial and terminal sides Measurement of angles: Degrees: 0 degrees make up one circle. Radians: one radian is the central angle formed by laying the radius of the circle onto the circumference. There are π radians in one circle. Revolutions: a full rotation of a circle. Conversion formula for angles: 0 " radians revolution Example ) Convert the following angles to the other two measurements Degrees Radians Revolutions a. 0 " b. " 0 c. 90 " d. 5 " e. " 5 f. 0 " g. 5" 5 h. 00 5" i. " j. 0 " " k. 0" " π. Radian and Degree Measurement - -

2 Conversion of angles expressed in degrees, minutes, and onds to decimal degrees: Example: Express 7 " " to decimal degrees: Example: Express.9 in degrees, minutes, onds:.9( 0) 55..( ). Example ) Convert from degrees, minutes, and onds to decimal degrees or vice versa. ( decimal places) Degrees, Minutes, Seconds Decimal Degrees a. 0 ".5 b. 5 " 5 ".9 c. 5 ".5 d. 9 5 " " 9. e " 59 " 0 f. 7 0 " " 7.00 Finding the arc length of a circle: We know that the circumference of a circle is given by C "r where r is the radius of the circle: This formula can also be used to find the length of an arc intercepted by some angle ". Arc length formula: s r" where r is measured in linear units, " is measured in radians, and s is measured in linear units Example : Find the arc length of the arc with radius inches and " 0 " & s in( 0 )% (.9 in. $ 0 ' Note: since the angle is measured in radians, it technically has no units so s is measured in linear units. Example : If the arc length is inches and the radius is inches, find the central angle in degrees " s r in in $ 0 ' & ) 7.7 % ( Example 5: If the arc length is meters and the central angle is 5, find the radius of the circle.. Radian and Degree Measurement - - r s " m $ 0 ' & ) 0.97 meters 5 % (

3 Example : Assuming the earth. is a sphere of radius,000 miles. Miami, Florida is at latitude 5 7 " 9 " N while Erie, Pennsylvania is at 7 " 5 " N and the cities are on the same meridian (one city lies due north of each other). Find the distance between the cities. " & s 000 m(.5 )% ( 0. miles $ 0 ' Imagine an object traveling along a circular arc. The element of time is now added to the equation. In order to do problems in such situations, we need to identify variables that can express certain information. Important variables for problems in which an object is moving along a circular arc Variable Name Given in Use in formulas Sample measures r Radius Linear units linear units inches,.5 feet radian " (theta) Angle Degrees, radians, revs Radians 5,.5! R,.75 revs s Arc length Linear units Linear units. ft, 5 cm t Time Time units Time units,.5 hrs v Linear velocity linear units s linear units time t time 5 ft,mph " (omega) Angular velocity angle time " radians 5 degrees,rpm t time Example 7) What variable are you being given (r, ", s, t, v, " ) " I make a U-turn with my car. t It takes 5 minutes to complete the exam r The spoke of a wheel is 5 inches s A circular track measures 00 feet " Around the world in 0 days v The space shuttle travels at,09 miles per hour Methods for transforming one variable into another You may multiply any variable by the fraction one. Here are some examples: inches 50 feet, foot mile, " radians rev, minute meters,000 0 km Example ): Convert the following: a. 5 miles to feet b. day to onds 50 ft 5 miles" 79,00 ft mile hr day " day c. 0,000 degrees to revolutions 0 feet d. " 00 hr to miles per hour 0000 " rev rev 0 ft " mile 00 " 50 ft hr e. 55 mph to feet f.,000,000,000 year to rpm,00. ft 55 mile hr " 50 ft mile " hr.7 ft 00,000,000,000 " rev year 0 " year 5 days " day 5.5 rev " 0 min min. Radian and Degree Measurement - -

4 The Angular velocity linear velocity formula: When an object is traveling along an arc, it has both an angular velocity and a linear velocity. The formula that ties these two variables together is: v "r or " v r " is always measured in radians time Examples 9: a. A bicycle s wheel has a 0 inch diameter. If the wheel makes.5 revolutions per ond, find the speed of the bike in mph. v "r.5 rev 5 in $ rev ft in mile 50 ft 00 hr.0 mph b. A flight simulator has pilots traveling in a circular path very quickly in order to experience g-forces. If the pilots are traveling at 00 mph and the circular room has a radius of 5 feet, find the number of rotations that simulator makes per ond. 00 miles " v r hr 5 ft 00 miles 50 ft hr 5 ft mile rev $ hr.75 rev 00 c. A large clock has its ond hand traveling at.5 inches per ond. Find the length of the ond hand..5 in r v " rev min.5 in min rev rev $ 0 min.7 in d. Two gears are connected by a belt. The large gear has a radius of inches while the small gear has a radius of inches. If a point on the small gear travels at rpm, find the angular velocity of the large gear. v w S r S v w L r L w S r S w L r L ( ) w L w L rpm. Radian and Degree Measurement - -

5 Unit - Radian and Degree Measure Homework. Convert the following angles to the other two measurements Degrees Radians Revolutions a. 70 " b. 5 " c. 0 " d. 0 " e. 5 " f. 00 5" 5 g. 5 7" 7 h. 0 " i. " 5 5 j. 5 " 7 k. 900& % ( $ " ' " l. 00 0" 5. Convert from degrees, minutes, and onds to decimal degrees or vice versa. ( decimal places) Degrees, Minutes, Seconds Decimal Degrees a. 5 5 " 5.5 b. " 5 ".0 c. " ". d. 5 ".9 e. 9 ".5 f. 5 0 " 7 " 5.50 g. 7 " " 7. h " 5 " Radian and Degree Measurement - 5 -

6 . Of the three variables r, ", and s, you will be given two of them. Find the third. Angles should be found in the units specified). Specify units for other variables. r " s a. inches 0.9 in b..5 ft 5.5 ft c.. meters.5.50 meters d..5 cm 5 " " 5.95 cm e.. miles.50. miles f mm mm g. 5 inches 75 " " feet h..97 in ".5 inches i. 7. ft ft. Convert the given quantity into the specified units. Show your work in the Convert to column. Given Convert to a..5 ft.5 ft in 5 in " ft b. 0 years 0 yrs 5 days hrs 00,5,0,000 " " " yr day hr c.,500 revolutions 500 rev 0 " rev 50,000 d. 0 km 0 km 0. ft,0 ft " km e.,500π 500! rev,500 rev " f. 5 ft 5 ft " mile mph " 50 ft hr g. rev rev min min " rev 0 min hr " "! hr day, day h. 500, week week " rev 0 " week 7 days " day hours " hour.0 rpm 0 min i. 0 mph 0 miles 50 ft inch hour,05 inch " " " hour miles ft 00 j. rev rev 0 days 0 days " 0 rev " day hours " hour 0 min 0.00 min. Radian and Degree Measurement - -

7 5. Find the distances between the cities with the given latitude, assuming that the earth is a sphere of radius,000 miles and the cities are on the same meridian. a. Dallas, Texas 7 " 9 " N and Omaha, Nebraska 5 " " N s r" 000 miles.7 $ 59 miles 0 b. San Francisco, California 7 " 9 " N and Seattle, Washington 7 " " N s r" 000 miles 9. $ miles 0 c. Copenhagen, Denmark 55 " " N and Rome, Italy 9 " " N s r" 000 miles.7 $ 959 miles 0 d. Jerusalem, Israel 7 " 0 " N and Johannesburg, South Africa 0 " S s r" 000 miles $,0 miles 0. What variable are you being given (r, ", s, t, v, " )? t a. It takes minutes to travel between classes. " b. It takes 5 minutes to walk around the school. " c. The Space Shuttle made 5 orbits of the earth. s d. The circumference of the orange is. inches. v e. The merry-go-round travels at a constant speed of miles per hour. " f. A Ferris-Wheel ride consists of revolutions. " g. That Ferris-Wheel completes the revolutions in minutes. r h. A propeller is 5 inches long. s i. The park is circular and I walked miles around its circumference. " j. An ant walking around a tire lying on the ground can only cover 5 degrees every minute. 7. Complete the chart, finding the missing information in the measurement requested. Show work. " r v Units Desired a. 0 rpm feet,005. ft feet/min b. 5 rev/.5 feet 0.5 mph mph c 55 o /. mile 0.7 mph mph d rpm foot 0 ft/min rpm e. 7.7 rpm 5 inches 0 mph rpm f..55 min miles 00 ft/ degrees/min g. 50 rpm.005 miles 00 mph miles h. 00 rev/.0 ft 50 feet/ feet i.,000 rev/.07 in 5,000 mph inches. Radian and Degree Measurement - 7 -

8 . Applications For each problem, draw a picture if necessary and show how you got your answer. a) A clock has a ond hand of length inches. How far in inches does the tip travel from when it is on the to when it is on the. s r" in 0 $.755 inches 0 b) The pendulum in the Franklin Institute is 0 feet long. It swing through an angle of o '. Find the length of the arc it swings through in inches. s r" 0 ft. $ in 95.5 inches 0 ft c) When the central angle is small and the distance to an object is large, the arc length formula is a good estimator of the height of the object. The angle of elevation of the Empire State Building from miles away is o '. Use the arc length formula to estimate its height in feet. s r" miles.7 $ 50 ft 55. ft 0 mile d) A car tire with radius inches rotates at rpm. Find the velocity of the car in mph. rev v "r min inch $ rev ft inch mile 50 ft 0 min hr.999 mph e) The Spinner is an amusement park ride that straps people to the edge of a circle and spins very fast. If riders are traveling at an actual speed of 5 mph, and the radius of the wheel is 5 feet, find the angular velocity of the wheel in rpm. " v r 5 miles hr 5 ft 50 ft mile rev $ hr. rpm 0 min. Radian and Degree Measurement - -

An angle on the coordinate plane is in standard form when the vertex is on the origin and one ray lies on the positive x-axis.

Name: Topic: Main Ideas/Questions Notes/Eamples Date: Class: Angles in Standard Form y θ An angle on the coordinate plane is in standard form when the verte is on the origin and one ray lies on the positive