Uniform Circular Motion

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Presentation transcript:

Uniform Circular Motion

Uniform Circular Motion 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! v = |v| = constant r r v  r

undergoes acceleration!! 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!!

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)] As Δt  0, Δv   v & Δv is in the radial direction  a  ac is radial!

a  ac is radial! Similar triangles  (Δv/v) ≈ (Δℓ/r) As Δt  0, Δθ  0, A B

(Δv/v) = (Δℓ/r)  Δv = (v/r)Δℓ Direction: Radially inward! 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: Radially inward! Centripetal  “Toward the center” Centripetal acceleration: Acceleration toward the center.

 ac  v always!! ac = (v2/r) Radially inward always! 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  v always!! ac = (v2/r)

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

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)

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

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)

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

Tangential & Radial acceleration Example 4.7: Over the Rise Problem: A car undergoes a constant acceleration of a = 0.3 m/s2 parallel 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?