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Circular Motion

This lecture covers topics such as uniform circular motion, centripetal acceleration, dynamics problems, angular acceleration, displacement, velocity, acceleration, and kinematics equations.

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Circular Motion

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  1. Lecture 08: Circular Motion • Uniform Circular Motion • Centripetal Acceleration • More Dynamics Problems • Circular Motion with Angular Acceleration • Displacement, Velocity, Acceleration • Kinematics Equations

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

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

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

  5. Driving Example • 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? • There are two forces on the car: • Normal • Gravity • Write F = ma: • FN – Fg = -m v2/R (note: acceleration is DOWN!) • FN – mg = -m v2/R • FN = 0 as you just barely leave the road… • -mg = -m v2/R • g = v2/R FN Fg v 18.8 m/s

  6. More Circular Motion(Non-Uniform) • Angular Displacement Dq = q2-q1 • How far (through what angle) it has rotated • Units: radians (2p radians = 1 revolution) • Angular Velocity w = Dq/Dt • How fast it is rotating • Units: radians/second • Angular Acceleration  = D/Dt • Change in angular velocity divided by time • Units: radians/second2 • Period = 1/frequency T = 1/f = 2p/w • Time to complete 1 revolution (or 2 radians) • Units: seconds

  7. Circular to Linear • Displacement Dx = R Dq(q in radians) • Velocity |v| = Dx/Dt = R Dq/Dt = Rw • Acceleration |a| = Dv/Dt = R Dw/Dt = R

  8. Kinematics for Circular Motion w/ constanta Linear Variables x,v,a (constant a). Angular Variables q,w,a (constant a).

  9. Gears Example • 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. • Note that 0 = 94 rad/s and  = 209 rad/s • Find angular acceleration: • Find angular displacement: • Find final linear speed:  = 230 rad/s2  = 76 rad v = 42 m/s

  10. Summary of Concepts • Uniform Circular Motion • Speed is constant • Direction is changing • Acceleration toward center a = v2 / R • Newton’s Second Law F = ma • Circular Motion with Angular Acceleration • q = angular position: rad. • w = angular velocity: rad/s • a = angular acceleration: rad/s2 • Linear to Circular conversions (x = Rq, v = Rw, a = Ra) • Kinematics Equations

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