1. Kinematics II Week 2

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

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

4. Quiz 1 Which of the following radian measures is equivalent to 90˚? 1.2π 2.π 3.π/2 4.π/3 5.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 When radius = 1 arc = Ө When radius = r When radius = r arc = rӨ * Ө must be in radians Ө arc r

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

8. Quiz 2 An object in circular motion moves 180˚ per second has an angular frequency of what? 1.2π 2.3π/2 3.π/2 4.π/3 5.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 ANSWER: ω = π/2 radians/sec

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

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

12. Quiz 3 An object in circular motion travels at 3 radians/sec in a circle of radius 5m. What is its angular speed? 1.3/5 m/sec 2.5/3 m/sec 3.2 m/sec 4.8 m/sec 5.15 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 An object travelling in circular motion experiences a centripetal force F The induced acceleration is The induced acceleration is a = v 2 /r F

16. Quiz 4 Imagine that the centripetal force applied on an object moving in circular motion is suddenly shut off. What would happen? 1.The object spirals outwards 2.The object spirals inwards 3.The object continues to move in uniform circular motion 4.The object decelerates while continuing to move in a circle 5.The object moves away in a straight line

17. 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)

18. Projectile Motion An object experiences gravitational acceleration g=9.80m/s 2 downwards as it moves An object experiences gravitational acceleration g=9.80m/s 2 downwards as it moves

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

20. x-direction The x-direction experiences NO acceleration, thus x = v x t (It’s that simple!)

21. y-direction The y-direction experiences an acceleration downwards, thus y = v y t – ½gt 2

22. Velocity Components x = v x t y = v y t – ½gt 2 v vxvxvxvx vyvyvyvy Ө v x = v cosӨ v y = v sinӨ

23. Quiz 5 At what angle should a cannon be positioned so that the projectile travels the farthest? 1.90˚ 2.60˚ 3.45˚ 4.30˚ 5.0˚

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

25. Proof of Quiz 5 x = v x t y = v y t – ½gt 2 v x = v cosӨ v y = v sinӨ Let y = 0  t = 2v y /g  x = 2v x v y /g  x = (2v 2 sinӨ cosӨ)/g  x = (v 2 sin2Ө)/g  x max = v 2 /g when 2Ө = 90˚  Ө = 45˚ y = 0 * Trig. Identity: sin2Ө = 2sinӨ cosӨ

26. Review Linear Motion under Uniform Acceleration Linear Motion under Uniform Acceleration x = x 0 + v 0 t + ½at 2 v = v 0 + at Δx = v ave t = [(v 0 + v)/2]t v 2 - v 0 2 = 2aΔx Uniform Circular Motion Ө = arc length (r=1) arc = rӨ ω = Ө/sec v = arc/sec = rω a = v 2 /r Projectile Motion x = v x t y = v y t – ½gt 2 v x = v cosӨ v y = v sinӨ

27. Next Week Choice 1 Choice 1 –Review kinematics Choice 2 Choice 2 –Newton’s laws

28. The End