Physics 101: Lecture 8, Pg 1 Centripetal Acceleration and Circular Motion Physics 101: Lecture 08 l Today’s lecture will cover Chapter 5 Exam II Problems Chapter 5: 1,5,13,39,47

Physics 101: Lecture 8, Pg 2 A B C Answer: B v Circular Motion Act 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? 10

Physics 101: Lecture 8, Pg 3 Acceleration in Uniform Circular Motion vv1vv1 vv2vv2 vvvv vv2vv2 vv1vv1 R RRRR centripetal acceleration a  v a ave =  v /  t Acceleration inward 12 Acceleration is due to change in direction, not speed. Since turns “toward” center, must be a force toward center.

Physics 101: Lecture 8, Pg 4 Preflights Consider the following situation: You are driving a car with constant speed around a horizontal circular track. On a piece of paper, draw a Free Body Diagram (FBD) for the car. How many forces are acting on the car? A) 1 B) 2 C) 3 D) 4 E) 5 f W FNFN correct 16  F = ma = mv 2 /R a=v 2 /R R 2% 23% 34% 28% 13% “Fn = Normal Force, W = Weight, the force of gravity, f = centripetal force.” “Gravity, Normal Force, Friction”

Physics 101: Lecture 8, Pg 5 Common Incorrect Forces Acceleration:  F = ma è Centripetal Acceleration l Force of Motion (Inertia not a force) è Forward Force, è Force of velocity è Speed l Centrifugal Force (No such thing!) è Centripetal (really acceleration) è Inward force (really friction) l Internal Forces (don’t count, cancel) è Car è Engine

Physics 101: Lecture 8, Pg 6 Preflights Consider the following situation: You are driving a car with constant speed around a horizontal circular track. On a piece of paper, draw a Free Body Diagram (FBD) for the car. The net force on the car is f W FNFN 16 A. Zero B. Pointing radially inward C. Pointing radially outward  F = ma = mv 2 /R a=v 2 /R R correct 12% 84% 5% “Ever spin stuff in a bowl? It gravitates outward.” “Centripetal force is always pointing inward" “This is why many racetracks have banked roadways”

Physics 101: Lecture 8, Pg 7ACT 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 F N exerted on you by the car seat as you drive past the bottom of the hill A. F N mg v mg FNFN R  F = ma F N - mg = mv 2 /R F N = mg + mv 2 /R a=v 2 /R correct 20

Physics 101: Lecture 8, Pg 8 Roller Coaster Example What is the minimum speed you must have at the top of a 20 meter roller coaster loop, to keep the wheels on the track. 25 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) = 9.9 m/s

Physics 101: Lecture 8, Pg 9 Circular Motion Angular displacement  =  2 -  1 è How far it has rotated Angular velocity  =  t è How fast it is rotating  Units radians/second 2  = 1 revolution Period =1/frequency T = 1/f = 2  è Time to complete 1 revolution 27

Physics 101: Lecture 8, Pg 10 Circular to Linear Displacement  s = r  in radians) Speed |v| =  s/  t = r  /  t = r  l Direction of v is tangent to circle 29

Physics 101: Lecture 8, Pg 11 l 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. è 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 Merry-Go-Round ACT 33

Physics 101: Lecture 8, Pg 12 Merry-Go-Round ACT II l Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go- round makes one complete revolution every two seconds. è Klyde’s angular velocity is: (a) (a) the same as Bonnie’s (b) (b) twice Bonnie’s (c) (c) half Bonnie’s Klyde Bonnie The angular velocity  of any point on a solid object rotating about a fixed axis is the same.  Both Bonnie & Klyde go around once (2  radians) every two seconds. 37

Physics 101: Lecture 8, Pg 13 Angular Acceleration Angular acceleration is the change in angular velocity  divided by the change in time. l If the speed of a roller coaster car is 15 m/s at the top of a 20 m loop, and 25 m/s at the bottom. What is the cars average angular acceleration if it takes 1.6 seconds to go from the top to the bottom? 41 = 0.64 rad/s 2

Physics 101: Lecture 8, Pg 14 Summary (with comparison to 1-D kinematics) AngularLinear And for a point at a distance R from the rotation axis: x = R  v =  R  a =  R

Physics 101: Lecture 8, Pg 15 CD Player Example l The CD in your disk player spins at about 20 radians/second. If it accelerates uniformly from rest with angular acceleration of 15 rad/s 2, how many revolutions does the disk make before it is at the proper speed? 48  = 13.3 radians 1 Revolutions = 2  radians  = 13.3 radians = 2.12 revolutions

Physics 101: Lecture 8, Pg 16 Summary of Concepts l Uniform Circular Motion è Speed is constant è Direction is changing è Acceleration toward center a = v 2 / r è Newton’s Second Law F = ma l Circular Motion   = angular position radians   = angular velocity radians/second   = angular acceleration radians/second 2  Linear to Circular conversions s = r  l Uniform Circular Acceleration Kinematics è Similar to linear! 50