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Chapter 5: Circular Motion; Gravitation

Chapter 5: Circular Motion; Gravitation. Sect. 5-1: Uniform Circular Motion. v = |v| = constant. Motion of a mass in a circle at constant speed . Constant speed  The Magnitude (size) of the velocity vector v is constant. BUT the DIRECTION of v changes continually !. r. r.

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Chapter 5: Circular Motion; Gravitation

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  1. Chapter 5: Circular Motion; Gravitation

  2. Sect. 5-1: Uniform Circular Motion v = |v| = constant • Motion of a mass in a circle at constant speed. • Constant speed  TheMagnitude(size) of the velocity vector v is constant. BUT the DIRECTION of v changes continually! r r v  r

  3. v = |v| = constant • The motion of a mass in a circle at Constant Speed. Is there an acceleration? To answer this, think about Newton’s 1st Law & Newton’s 2nd Law! r r

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

  5. Look at the velocity change Δv in the limit that the time interval Δt becomes infinitesimally small & get: Centripetal (Radial) Acceleration Similar triangles  (Δv/v) ≈ (Δℓ/r) AsΔt  0, Δθ  0, A B AsΔt  0, Δv   v & Δv is in the radial direction  a  aRis radial!

  6. (Δv/v) = (Δℓ/r) Δv = (v/r)Δℓ • Note that the acceleration (radial) is aR = (Δv/Δt) =(v/r)(Δℓ/Δt) As Δt  0, (Δℓ/Δt) v Magnitude: 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 = aR: Centripetal acceleration. Radially inward always!  aR valways!!

  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 the 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: aR = (v2/r) = (4π2r/T2)

  10. Example 5-1: Acceleration of a revolving ball A 150-g ball at the end of a string is revolving uniformly in a horizontal circle of radius 0.600 m. The ball makes 2.00 revolutions per second. Calculate its centripetal acceleration. r r

  11. Example 5-2: Moon’s Centripetal Acceleration The Moon’s nearly circular orbit about the Earth has a radius of about 384,000 km and a period T of 27.3 days. Calculate the acceleration of the Moon toward the Earth.

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