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Kinematics II

Kinematics II. Week 2. Contents. Radians Uniform Circular Motion Projectile Motion Summary of Kinematics I & II. A radian is a measure of an angle. Radius = 1 Ө = arc length 360˚ = 2 π radians 180 ˚ = π radians. Radians. arc. Ө. Unit Circle; radius = 1. Quiz 1.

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Kinematics II

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  1. Kinematics II Week 2

  2. Contents • Radians • Uniform Circular Motion • Projectile Motion • Summary of Kinematics I & II

  3. A radian is a measure of an angle. Radius = 1 Ө = arc length 360˚ = 2π radians 180˚ = π radians Radians arc Ө Unit Circle; radius = 1

  4. Quiz 1 Which of the following radian measures is equivalent to 90˚? • 2π • π • π/2 • π/3 • π/4

  5. Quiz 1 Which of the following radian measures is equivalent to 90˚? 180˚ = π radians  90˚ = π/2 radians ANSWER: π/2

  6. Arc Length • When radius = 1 arc = Ө • When radius = r arc = rӨ * Ө must be in radians arc r Ө

  7. Uniform Circular Motion • “Uniform” implies constant speed while moving in a circle • Angular Frequency • ω = Ө/sec Ө

  8. Quiz 2 An object in circular motion moves 180˚ per second has an angular frequency of what? • 2π • 3π/2 • π/2 • π/3 • π/4

  9. Quiz 2 An object in circular motion moves 180˚ per second has an angular frequency of what? ANSWER: ω = π/2 radians/sec

  10. Angular Speed arc = rӨ ω = Ө/sec • Angular speed: v = arc/sec = (rӨ)/sec = r(Ө/sec) = rω Ө Ө

  11. Summary • Ө = arc length (r=1) • arc = rӨ • ω = Ө/sec • v = arc/sec = rω arc Ө r

  12. Quiz 3 An object in circular motion travels at 3 radians/sec in a circle of radius 5m. What is its angular speed? • 3/5 m/sec • 5/3 m/sec • 2 m/sec • 8 m/sec • 15 m/sec

  13. Quiz 3 An object in circular motion travels at 3 radians/sec in a circle of radius 5m. What is its angular speed? ω = 3 radians/sec v = rω v = (5)(3) = 15 m/sec ANSWER: 15 m/sec

  14. Centripital Force • An object travelling in circular motion experiences a centripetal force F • The induced acceleration is a = v2/r F

  15. Quiz 4 Imagine that the centripetal force applied on an object moving in circular motion is suddenly shut off. What would happen? • The object spirals outwards • The object spirals inwards • The object continues to move in uniform circular motion • The object decelerates while continuing to move in a circle • The object moves away in a straight line

  16. Quiz 4 Imagine that the centripetal force applied on an object moving in circular motion is suddenly shut off. What would happen? ANSWER: The object moves away in a straight line (Newton’s first law)

  17. Projectile Motion • An object experiences gravitational acceleration g=9.80m/s2 downwards as it moves

  18. Projectile Motion • Projectile motion is really two independent 1-D motions • One in the x direction • One in the y direction

  19. x-direction The x-direction experiences NO acceleration, thus x = vxt (It’s that simple!)

  20. y-direction The y-direction experiences an acceleration downwards, thus y = vyt – ½gt2

  21. Velocity Components x = vxt y = vyt – ½gt2 vx = v cosӨ vy= v sinӨ v vy Ө vx

  22. Quiz 5 At what angle should a cannon be positioned so that the projectile travels the farthest? • 90˚ • 60˚ • 45˚ • 30˚ • 0˚

  23. Quiz 5 At what angle should a cannon be positioned so that the projectile travels the farthest? ANSWER: 45˚

  24. Proof of Quiz 5 x = vxt y = vyt – ½gt2 vx = v cosӨ vy= v sinӨ Let y = 0  t = 2vy/g  x = 2vxvy/g  x = (2v2 sinӨ cosӨ)/g  x = (v2 sin2Ө)/g  xmax = v2/g when 2Ө = 90˚  Ө = 45˚ y = 0 y = 0 * Trig. Identity: sin2Ө = 2sinӨ cosӨ

  25. Linear Motion under Uniform Acceleration x = x0 + v0t + ½at2 v = v0 + at Δx = vavet = [(v0 + v)/2]t v2 - v02 = 2aΔx Uniform Circular Motion Ө = arc length (r=1) arc = rӨ ω = Ө/sec v = arc/sec = rω a = v2/r Projectile Motion x = vxt y = vyt – ½gt2 vx = v cosӨ vy= v sinӨ Review

  26. Next Week • Choice 1 • Review kinematics • Choice 2 • Newton’s laws

  27. The End

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