1. Do now:Hātarei, 24 Whiringa-ā-nuku 2015 A mass oscillates with SHM, period T = 2.4 s and amplitude A = 0.12m 1)Use trig to calculate the angle  2)Calculate the time taken to travel from point C to B 3)Calculate the time taken to travel from point B to A

2. Quickly try:Hātarei, 24 Whiringa-ā-nuku 2015 A mass oscillates with SHM, period T = 2.4 s and amplitude A = 0.12m 4) Where is the mass at t = 1.0s? 5) What is the velocity of the mass at t = 1.0s? 6) What is the acceleration of the mass at t = 1.0s?

3. More practise:Hātarei, 24 Whiringa-ā-nuku 2015 A mass oscillates with SHM, period T = 2.4 s and amplitude A = 0.12m 7) Where is the mass at t = 1.8s? 8) What is the velocity of the mass at t = 1.8s? 9) What is the acceleration of the mass at t = 1.8s?

4. 4 SHM Velocity For p and m to be staying in line… For p and m to be staying in line… …they must both have the same vertical component to their speeds at any time. …they must both have the same vertical component to their speeds at any time. p m  A vmvm vpvp What is the relationship between the SHM velocity v m and the velocity of the test particle v p on the reference circle? What is the relationship between the SHM velocity v m and the velocity of the test particle v p on the reference circle?

5. So substituting this back into the first equation: So substituting this back into the first equation: SHM Velocity v m must be equal to the vertical component of v p : v m must be equal to the vertical component of v p : vmvm vpvp  The reference circle has constant v p : The reference circle has constant v p : This formula will be given in the exam. Extension: derive using calculus

6. 6Displacement-time A graph of displacement-time can be drawn by using a rotating radius vector known as a Phasor. A graph of displacement-time can be drawn by using a rotating radius vector known as a Phasor. t ω t=0 t=1 t=2 t=3 t=5 t=6 t=7 17235468

7. 7 Sketch the velocity-time graph for SHM t ω t=6 t=7 t=0 t=1 t=3 t=4 t=5 17235468 +Aω – Aω Velocity Watch video clip

8. 8 Velocity Phasor t ω 17235468 velocity displacement The length of the velocity phasor is…The length of the velocity phasor is… What is the phase difference between the velocity phasor and the diplacement phasor?What is the phase difference between the velocity phasor and the diplacement phasor? The velocity phasor is 90° (or  /2 rad) ahead of the displacement phasor  A

9. 9 SHM Acceleration Point p has centripetal acceleration a p in towards the centre of the circle Point p has centripetal acceleration a p in towards the centre of the circle The SHM object has acceleration a m towards equilibrium... The SHM object has acceleration a m towards equilibrium......which must be equal to the vertical component of a p...which must be equal to the vertical component of a p p m  amam apap

10. 10 SHM Acceleration amam apap  Negative sign indicates direction is always towards equilibrium But a p is centripetal, so: But a p is centripetal, so: Putting this all together: Putting this all together:

11. 11 t t=5 ω t=0 t=6 t=7 t=1 t=2 t=3 17235468 Sketch the acceleration-time graph for SHM

12. 12 All three…. t ω 17235468 velocity displacement acceleration Clip: Spring SHM with d v and a graphs alongside - 20sec

13. 13 All three… If t=0 is not equilibrium, then all the phasors shift their starting positions. The graphs start in different positions and the sin/cosine function may change. If t=0 is not equilibrium, then all the phasors shift their starting positions. The graphs start in different positions and the sin/cosine function may change. Do pg 124 Qu 13-22

14. 14 Acceleration