Circular Motion Lecture 08: l Uniform Circular Motion è Centripetal Acceleration è More Dynamics Problems l Circular Motion with Angular Acceleration è Displacement, Velocity, Acceleration è Kinematics Equations

v Uniform Circular Motion An object moving in a circle with constant velocity.

Acceleration in Uniform Circular Motion l Centripetal Acceleration è Due to change in DIRECTION (not speed) è Direction of Acceleration: INWARD è Magnitude of Acceleration: v R

Uniform Circular Motion v R Instantaneous velocity is tangent to circle. Instantaneous acceleration is radially inward. There must be a net inward force to provide the acceleration. a

Driving Example l As you drive over the top of a hill (with radius of curvature of 36 m) in your minivan, at what speed will you begin to leave the road? è There are two forces on the car: »Normal »Gravity è Write  F = ma: »F N – F g = -m v 2 /R (note: acceleration is DOWN!) »F N – mg = -m v 2 /R è F N = 0 as you just barely leave the road… »-mg = -m v 2 /R »g = v 2 /R v FNFN FgFg 18.8 m/s

More Circular Motion (Non-Uniform) Angular Displacement  =  2 -  1 è How far (through what angle) it has rotated  Units: radians (2  radians = 1 revolution) Angular Velocity  =  t è How fast it is rotating è Units: radians/second Angular Acceleration  =  t è Change in angular velocity divided by time è Units: radians/second 2 Period = 1/frequency T = 1/f = 2  è Time to complete 1 revolution (or 2  radians) è Units: seconds

Circular to Linear Displacement  x = R  in radians) Velocity |v| =  x/  t = R  /  t = R  Acceleration |a| =  v/  t = R  /  t = R 

Kinematics for Circular Motion w/ constant  Linear Variables x,v,a (constant a). Angular Variables , ,  (constant 

Gears Example l One of the gears in your car has a radius of 20 cm. Starting from rest it accelerates from 900 rpm to 2000 rpm in 0.5 s (rpm stands for revolutions per minute). Find the angular acceleration, the angular displacement during this time, and the final linear speed of a point on the outside of the gear. è Note that  0 = 94 rad/s and  = 209 rad/s è Find angular acceleration: è Find angular displacement: è Find final linear speed:  = 230 rad/s 2  = 76 rad v = 42 m/s

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 with Angular Acceleration   = angular position: rad.   = angular velocity: rad/s   = angular acceleration: rad/s 2  Linear to Circular conversions (x = R  v = R  a = R  è Kinematics Equations

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