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Do now! DEFINITIONS TEST!! You have 12 minutes!. Topic 4 Oscillations and Waves.

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Presentation on theme: "Do now! DEFINITIONS TEST!! You have 12 minutes!. Topic 4 Oscillations and Waves."— Presentation transcript:

1 Do now! DEFINITIONS TEST!! You have 12 minutes!

2 Topic 4 Oscillations and Waves

3 Aims Remember the terms displacement, amplitude, frequency, period and phase difference. Define simple harmonic motion (a = -ω 2 x) Solve problems using a = -ω 2 x Apply the equations x = x 0 cosωt, x = x 0 sinωt, v = v 0 sinωt, v = v 0 cosωt, and v = ±ω(x 0 2 – x 2 )

4 Displacement - x The distance and direction from the equilibrium position. = displacement

5 Amplitude - A The maximum displacement from the equilibrium position. amplitude

6 Period - T The time taken (in seconds) for one complete oscillation. It is also the time taken for a complete wave to pass a given point. One complete wave

7 Frequency - f The number of oscillations in one second. Measured in Hertz. 50 Hz = 50 vibrations/waves/oscillations in one second.

8 Period and frequency Period and frequency are reciprocals of each other f = 1/TT = 1/f

9 Phase difference is the time difference or phase angle by which one wave/oscillation leads or lags another. 180° or π radians

10 Phase difference is the time difference or phase angle by which one wave/oscillation leads or lags another. 90° or π/2 radians

11 Simple harmonic motion (SHM) periodic motion in which the restoring force is proportional and in the opposite direction to the displacement

12 Hookes law What can you remember?

13 Simple harmonic motion (SHM) periodic motion in which the restoring force is in the opposite durection and proportional to the displacement F = -kx

14 Graph of motion A graph of the motion will have this form Time displacement

15 Graph of motion A graph of the motion will have this form Time displacement Amplitude x 0 Period T

16 Graph of motion Notice the similarity with a sine curve angle 2π radians π/2 π 3π/2 2π

17 Graph of motion Notice the similarity with a sine curve angle 2π radians π/2 π 3π/2 2π Amplitude x 0 x = x 0 sinθ

18 Graph of motion Time displacement Amplitude x 0 Period T

19 Graph of motion Time displacement Amplitude x 0 Period T x = x 0 sinωt where ω = 2π/T = 2πf = (angular frequency in rad.s -1 )

20 When x = 0 at t = 0 Time displacement Amplitude x 0 Period T x = x 0 sinωt where ω = 2π/T = 2πf = (angular frequency in rad.s -1 )

21 When x = x 0 at t = 0 Time displacement Amplitude x 0 Period T x = x 0 cosωt where ω = 2π/T = 2πf = (angular frequency in rad.s -1 )

22 When x = 0 at t = 0 Time displacement Amplitude x 0 Period T x = x 0 sinωt v = v 0 cosωt where ω = 2π/T = 2πf = (angular frequency in rad.s -1 )

23 When x = x 0 at t = 0 Time displacement Amplitude x 0 Period T x = x 0 cosωt v = -v 0 sinωt where ω = 2π/T = 2πf = (angular frequency in rad.s -1 )

24 To summarise! When x = 0 at t = 0 x = x 0 sinωt and v = v 0 cosωt When x = x 0 at t = 0 x = x 0 cosωt and v = -v 0 sinωt It can also be shown that v = ±ω(x 0 2 – x 2 ) and a = -ω 2 x where ω = 2π/T = 2πf = (angular frequency in rad.s -1 )

25 Maximum velocity?

26 When x = 0 At this point the acceleration is zero (no resultant force at the equilibrium position).

27 Maximum acceleration?

28 When x = +/– x 0 Here the velocity is zero

29 Oscillating spring We know that F = -kx and that for SHM, a = -ω 2 x (so F = -mω 2 x) So -kx = -mω 2 x k = mω 2 ω = (k/m) Remembering that ω = 2π/T T = 2π(m/k)

30 Lets do a simple practical! T = 2π(m/k)

31 We need to try some examples!

32 Example 1 Find the acceleration of a system oscillating with SHM where ω = 2.5 rad s -1 and x = 0.5 m to 2.s.f.

33 Example 1 Find the acceleration of a system oscillating with SHM where ω = 2.5 rad s -1 and x = 0.5 m to 2.s.f. Using a = -ω 2 x a = -(2.5) a = m.s -2

34 Example 2 Find the displacement at a point to 2.s.f. of a system of frequency 4 Hz when its acceleration is -8 ms -2.

35 Example 2 Find the displacement at a point to 2.s.f. of a system of frequency 4 Hz when its acceleration is -8 ms -2. ω = 2πf = 2π x 4 = 8π a = -ω 2 x x = -a/ω 2 = 8/(8π) 2 = 1/8π 2 = m

36 Example 3 For a SHM system of x = 0 at t = 0, find x when ω = 5.0 rad s -1, x o = 0.5 m and t = 1.0 s. What is the maximum acceleration of this system to 3.s.f? What is the maximum velocity of this system?

37 Example 3 For a SHM system of x = 0 at t = 0, find x when ω = 5.0 rad s -1, x o = 0.5 m and t = 1.0 s. What is the maximum acceleration of this system to 3.s.f? What is the maximum velocity of this system? x = x o sinωt (when x = 0 at t = 0) x = 0.5sin(5.0 x 1.0) = 0.5sin5 = m

38 Example 3 For a SHM system of x = 0 at t = 0, find x when ω = 5.0 rad s -1, x o = 0.5 m and t = 1.0 s. What is the maximum acceleration of this system to 3.s.f? What is the maximum velocity of this system? a = -ω 2 x Maximum acceleration when x = ±x o a max = -ω 2 x o = -(5) 2 x 0.5 = m.s -2

39 Example 3 For a SHM system of x = 0 at t = 0, find x when ω = 5.0 rad s -1, x o = 0.5 m and t = 1.0 s. What is the maximum acceleration of this system to 3.s.f? What is the maximum velocity of this system? v = ±ω(x 0 2 – x 2 ) Maximum velocity when x = 0 v max = ± ω(x 0 2 – x 2 ) = ±5.0(0.5) 2 = ±2.5 m.s -1

40 Lets try some questions! Finish for homework. Due Thursday 26 th February (two days before Mr Porters virtual birthday)


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