Lecture 08 Circular Motion l Uniform Circular Motion

Lecture 08: Circular Motion l Uniform Circular Motion èCentripetal Acceleration èMore Dynamics Problems l Circular Motion with Angular Acceleration èDisplacement, Velocity, Acceleration èKinematics Equations

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

Acceleration in Uniform Circular Motion R l Centripetal Acceleration èDue to change in DIRECTION (not speed) èDirection of Acceleration: INWARD èMagnitude of Acceleration: v

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

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? èThere are two forces on the car: » Normal » Gravity FN Fg èWrite F = ma: » FN – Fg = -m v 2/R (note: acceleration is DOWN!) » FN – mg = -m v 2/R èFN = 0 as you just barely leave the road… » -mg = -m v 2/R » g = v 2/R 18. 8 m/s v

More Circular Motion (Non-Uniform) l Angular Displacement D = 2 - 1 èHow far (through what angle) it has rotated èUnits: radians (2 radians = 1 revolution) l Angular Velocity = D /Dt èHow fast it is rotating èUnits: radians/second l Angular Acceleration = D /Dt èChange in angular velocity divided by time èUnits: radians/second 2 l Period = 1/frequency T = 1/f = 2 / èTime to complete 1 revolution (or 2 radians) èUnits: seconds

Circular to Linear l Displacement l Velocity Dx = R D ( in radians) |v| = Dx/Dt = R D /Dt = R l Acceleration |a| = Dv/Dt = R D /Dt = R

Kinematics for Circular Motion w/ constant a 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. èNote that 0 = 94 rad/s and = 209 rad/s èFind angular acceleration: = 230 rad/s 2 èFind angular displacement: = 76 rad èFind final linear speed: v = 42 m/s

Summary of Concepts l 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 Circular Motion with Angular Acceleration è = angular position: rad. è = angular velocity: rad/s è = angular acceleration: rad/s 2 è Linear to Circular conversions (x = R , v = R , a = R ) è Kinematics Equations

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