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Uniform Circular Motion

Uniform Circular Motion. Uniform Circular Motion. v = |v| = constant. Motion of a mass in a circle at constant speed . Constant speed  Magnitude (size) of velocity vector v is constant. BUT the DIRECTION of v changes continually!. r. r. v  r.

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Uniform Circular Motion

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  1. Uniform Circular Motion

  2. Uniform Circular Motion v = |v| = constant • Motion of a mass in a circle at constant speed. • Constant speed  Magnitude (size) of velocity vector v is constant. BUT the DIRECTION of v changes continually! r r v  r

  3. Mass moving in circle at constant speed. • Acceleration  Rate of change of velocity a = (Δv/Δt) Constant speed  Magnitude(size) of velocity vector v is constant. v = |v| = constant • BUT the DIRECTION of v changes continually! An object moving in a circle undergoes acceleration!!

  4. Centripetal (Radial) Acceleration Motion in a circle from A to point B. Velocity v is tangent to the circle.a = limΔt 0 (Δv/Δt)= limΔt 0[(v2 - v1)/(Δt)] AsΔt  0, Δv   v &Δv is in the radial direction  a  acis radial!

  5. a  acis radial! Similar triangles (Δv/v) ≈ (Δℓ/r) AsΔt  0, Δθ  0, A B

  6. (Δv/v) = (Δℓ/r) Δv = (v/r)Δℓ • Note that the acceleration (radial) is ac = (Δv/Δt) =(v/r)(Δℓ/Δt) As Δt  0, (Δℓ/Δt) v and Magnitude: ac = (v2/r) Direction: Radiallyinward! • Centripetal  “Toward the center” • Centripetal acceleration: Acceleration towardthe center.

  7. Typical figure for particle moving in uniform circular motion, radius r (speed v = constant): v :Tangent to the circle always! a = ac: Centripetal acceleration. Radially inward always!  ac valways!! ac= (v2/r)

  8. Period & Frequency • A particle moving in uniform circular motion of radius r (speed v = constant) • Description in terms of period T & frequencyf: • Period T time for one revolution (time to go around once), usually in seconds. • Frequency f  number of revolutions per second.  T = (1/f)

  9. Particle moving in uniform circular motion, radius r (speed v = constant) • Circumference = distance around= 2πr  Speed: v = (2πr/T) = 2πrf Centripetal acceleration: ac = (v2/r) = (4π2r/T2)

  10. Example 4.6: Centripetal Acceleration of the Earth Problem 1: Calculate the centripetal acceleration of the Earth as it moves in its orbit around the Sun. NOTE:The radius of the Earth's orbit (from a table) is r ~ 1.5  1011 m Problem 2: Calculate the centripetal acceleration due to the rotation of the Earth relative to the poles & compare with geo-stationary satellites. NOTE:From a table, the radius of the Earth is r ~ 6.4  106 m & the height of geo-stationary satellites is ~ 3.6  107 m

  11. Tangential & Radial Acceleration • Consider an object moving in a curved path. If the speed |v| of object is changing, there is an acceleration in the direction of motion. ≡ Tangential Acceleration at ≡ |dv/dt| • But, there is also always a radial (or centripetal) acceleration perpendicular to the direction of motion. ≡ Radial (Centripetal) Acceleration ar = |ac| ≡ (v2/r)

  12. Total Acceleration  In this case there are always two vector components of the acceleration: • Tangential:at = |dv/dt| &Radial:ar = (v2/r) • Total Acceleration is the vector sum: a = aR+ at

  13. Tangential & Radial acceleration Example 4.7: Over the Rise Problem:A car undergoes a constant acceleration of a = 0.3 m/s2parallel to the roadway. It passes over a rise in the roadway such that the top of the rise is shaped like a circle of radius r = 515 m. At the moment it is at the top of the rise, its velocity vector v is horizontal & has a magnitude of v = 6.0 m/s. What is the direction of the total acceleration vector a for the car at this instant?

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