1. Lecture 7Purdue University, Physics 2201 UNIMPORTABLE: #817EE11E, 4.00 #8279AE55, 2.00 #834C955A, 4.00 #83CA7831, 4.00 #841D4BD2,4.00

2. Lecture 7Purdue University, Physics 2202 Lecture 07 Circular Motion PHYSICS 220

3. Lecture 7Purdue University, Physics 2203 Examples of Circular Motion

4. Uniform Circular Motion Assume constant speed The direction of the velocity is continually changing –The vector is always tangent to the circle Uniform circular motion assumes constant speed period of the motion: T = 2  r/v [s]

5. Lecture 7Purdue University, Physics 2205 Angular Variables The motion of objects moving in circular (or nearly circular) paths, is often described by angles measured in radians rather than degrees. The angle  in radians, is defined as: If s = r the angle is 1 rad If s = 2  r (the circumference of the circle) the angle is 2  rad. (In other words, 360° = 2  rad.)

6. Lecture 7Purdue University, Physics 2206 Circular Motion Period = 1/frequency T = 1/f Time to complete 1 revolution Angular displacement  =  2 -  1 How far it has rotated Angular velocity  av =  t How fast it is rotating Units: radians/second (2  = 1 revolution)  2 

7. Lecture 7Purdue University, Physics 2207 Circular to Linear Displacement  s = r  in radians) Speed |v| = |  s/  t| = r |  /  t| = r|  |v| = 2  rf |v| = 2  r/T Direction of v is tangent to circle

8. Lecture 7Purdue University, Physics 2208 v = r  = 0.06  20 = 1.2 m/s The speed could be obtained by |v| = 2  r/T A CD spins with an angular velocity 20 radians/second. What is the linear speed 6 cm from the center of the CD? Examples The wheel of a car has a radius of 0.29 m and is being rotated at 830 revolution per minute (rpm) on a tire- balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving:

9. Lecture 7Purdue University, Physics 2209 Bonnie sits on the outer rim of a merry-go-round with radius 3 meters, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds. Klyde’s speed is: A) A) the same as Bonnie’s B) B) twice Bonnie’s C) C) half Bonnie’s Klyde Bonnie Bonnie travels 2  R in 2 seconds v B = 2  R / 2 = 9.42 m/s Klyde travels 2  (R/2) in 2 seconds v K = 2  (R/2) / 2 = 4.71 m/s iClicker

10. Lecture 7Purdue University, Physics 22010 Centripetal Acceleration Magnitude of the velocity vector is constant, but direction is constantly changing At any instant of time, the direction of the instantaneous velocity is tangent to the path Therefore: nonzero acceleration

11. Lecture 7Purdue University, Physics 22011 Uniform Circular Motion v R Instantaneous velocity is tangent to circle Instantaneous acceleration is radially inward There must be a force to provide the acceleration a centripetal acceleration Circular motion with constant speed Recall: v =  R

12. Lecture 7Purdue University, Physics 22012 Answer: 2 Circular Motion A ball is going around in a circle attached to a string. If the string breaks at the instant shown, which path will the ball follow? 1 2 3 v 4 5

13. Lecture 7Purdue University, Physics 22013 Artificial Gravity

14. Lecture 7Purdue University, Physics 22014 Roller Coaster Example mg N y-direction: F = ma N + mg = m a Let N = 0, just touching mg = m a mg = m v 2 /R g = v 2 / R v = sqrt(g*R) = 10 m/s What is the minimum speed you must have at the top of a 20 meter diameter roller coaster loop, to keep the wheels on the track. y

15. Lecture 7Purdue University, Physics 22015 Unbanked Curve What force accelerates a car around a turn on a level road at constant speed? A) it is not accelerating B) the road on the tires C) the tires on the road D) the engine on the tires

16. Lecture 7Purdue University, Physics 22016 Unbanked Curve What is the maximum velocity a car can go around an unbanked curve in a circle without slipping? The maximum velocity to go around an un- banked curve depends only on  s (for a given r) Dry road:  s =0.9 Icy road:  s =0.1

17. Lecture 7Purdue University, Physics 22017 Banked Curve A car drives around a curve with radius 410 m at a speed of 32 m/s. The road is banked at 5.0°. The mass of the car is 1400 kg. A) What is the frictional force on the car? B) At what speed could you drive around this curve so that the force of friction is zero?

18. Lecture 7Purdue University, Physics 22018 y-direction x-direction  x y N W f Banked Curve (1) (2)

19. Lecture 7Purdue University, Physics 22019 2 equations and 2 unknown we can solve for N in (1) and substitute in (2) Banked Curve

20. Lecture 7Purdue University, Physics 22020 Banked Curve A car drives around a curve with radius 410 m at a speed of 32 m/s. The road is banked at 5.0°. The mass of the car is 1400 kg. A) What is the frictional force on the car? B) At what speed could you drive around this curve so that the force of friction is zero?

21. Lecture 7Purdue University, Physics 22021 Suppose you are driving through a valley whose bottom has a circular shape. If your mass is m, what is the magnitude of the normal force N exerted on you by the car seat as you drive past the bottom of the hill. A) N < mg B) N = mg C) N > mg v mg N R  F = ma N - mg = mv 2 /R N = mg + mv 2 /R a=v 2 /R correct iClicker Since there is centripetal acceleration, the normal force is greater than simply mg