Oscillations

2. A laser beam is shown into a piece of glass at an angle of 35º relative to the surface. What is the angle of refraction? • After it enters the glass it leave and enters a puddle of water below. What is the velocity of the light inside the puddle?

3. Oscillations 4.1.1 Describe examples of oscillations. • A periodic motion is one during which a body continually retraces its path at equal intervals

4. Oscillations • It’s motion is continually changing, periodically, such that it reaches it’s maximum and minimum positions returning to its original position in regular time periods. • Examples: child’s swing, a person bouncing on a trampoline, bungee jumper, springboard’s vibration after the diver has left the board, electrons in an antenna

5. P Oscillations • Displacement (s) is the instantaneous distance of the moving object from its mean position (in a specified direction). • A is the amplitude of motion: the distance from the centre of motion to either extreme • T is the period of motion: the time for one complete cycle of the motion O B A

6. Amplitude Oscillations 4.1.2 Define the terms displacement, amplitude, frequency, period and phase difference. The connection between frequency and period should be known. • Which ball has a larger amplitude? (Reminder: use student for demo) Ball A • Which ball has the larger period? Ball A

7. Frequency Oscillations • The frequency of motion, f, is the rate of repetition of the motion -- the number of cycles per unit time. There is a simple relation between frequency and period: • If ball B has a time period of 12 s, what is the frequency? f = 0.0833 Hz

8. Angular frequency Oscillations • Angular frequency is the rotational analogy to frequency. Represented as ω(omega) , and is the rate of change of an angle when something is moving in a circular orbit. This is the usual frequency (measured in cycles per second), converted to radians per second. That is • Which ball has the larger angular frequency? Ball B

9. Oscillations • Displayed below is a position-time graph of a piston moving in and out. Find the: Amplitude Period Frequency Angular frequency 10 cm 0.2 s 5.0 Hz 10p rads-1

10. Phase • In-phase – time for two particles to reach max and min is equal • Out-of-phase- time for two particles to reach max and min is NOT equal

11. Phase • Measured in either degree or radians • In-phase – exactly one time period apart, one full wavelength, 360º or 2π radians • Out-of-phase – not full period apart, not a full wavelength,

12. Roundabout • Child appears to be moving back and forth or in simple harmonic motion.

13. SHM • Characterized by an acceleration vector always directed toward the equilibrium position and directly proportional to the (-)displacement. • ω = 2π / T

14. Phase Difference • The angle by which one oscillation lags behind or leads in front of another oscillation. • a = -ω2x • a = aceeleration • ω = angular velocity • x = radius of circular motion

15. Practice Questions 4.1 • The diagram below shows a mass M suspended from a vertically supported spring. The mass is pulled down to the position marked A and released such that it oscillates with SHM between the positions A and B. The equilibrium poition of the mass is at the labeled position E. • The time period of the oscillation is 0.8s with an max acceleration of 4m/s2. • Draw a graph that shows how the acceleration of the mass varies with time over one time period. • Mark on the graph all the points that correspond to the positions A, B, and E on the diagram.

16. Practice Question 4.2 • A particle is undergoing simple harmonic motion. When it is passing through its equilibrium position, which one of the following about it’s acceleration and kinetic energy is correct? • A. zero accel. and max KE • B. zero accel. and zero KE • C. max accel. and zero KE • D. max accel. and max KE

17. Practice Question 4.3 • A mass on the end of a spring undergoes simple harmonic motion about an equilibrium position as shown. (Have Mr. B draw it.) • If the upward direction is taken as positive, which graph best represents how the acceleration of the mass varies with displacement from the equilibrium position? (Have Mr. B draw it.)

18. Practice Question 4.4 • When an object undergoes SHM, which of the following is true of the magnitude of the acceleration of the object? • A. It is uniform throughout the motion? • B. It is greatest at the end points of the motion. • C. It is greatest at the midpoint of the motion. • D. It is greatest at the midpoints and the endpoints.

19. Practice Questions

20. At the equilibrium position (x=0) with the mass moving upwards, it has it’s maximum velocity in the upwards (positive) direction. At the highest point, the velocity is zero, but the mass feels its maximum downward acceleration. In the lowest point, the velocity is zero again and the mass feels it’s maximum upward acceleration.

21. Phase Oscillations • Here is an oscillating ball. • Its motion can be described as follows: • Then it moves with v < 0 through the center to the left • Then it is at v = 0 at the left • Then it moves with v > 0 through the center to the right • Then it repeats...

22. Simple Harmonic Motion 4.1.3 Define simple harmonic motion (SHM) and state the defining equation as a = −ω2x . • Simple harmonic motion is defined as the motion that takes place when the acceleration, a, of an object is always directed towards, and proportional to, its displacement from a fixed point. This acceleration is caused by a restoring force that must always be pointed towards the mean position and also proportional to the displacement from the mean position. • F -X or F=-(constant) x X • Since F=ma • a -X or a=-(constant) x X • The negative sign signifies that the acceleration is always pointing back towards the mean position.

23. Simple Harmonic Motion • The constant of proportionality between acceleration and displacement is often identified as the square of a constant ω which is referred to as the angular frequency. • a=-ω2x Acceleration a / ms-2 Gradient of line = -ω2 Displacement x / m -A A

24. Simple Harmonic Motion • Points to note about Simple Harmonic Motion: • The time period T does not depend on the amplitude A. • Not all oscillations are simple harmonic motion, but there are many everyday examples of natural simple harmonic motion.

25. Oscillations • Watch the oscillating duck. Let's consider velocity now • Remember that velocity is a vector, and so has both negative and positive values. • Where does the magnitude of v(t) have a maximum value? C • Where does v(t) = 0? A and E

26. Oscillations • Watch the oscillating duck. Let's consider acceleration now • Remember that acceleration is a vector, and so has both negative and positive values. • Where does the magnitude of a(t) have a maximum value? A and E • Where does a(t) = 0? C

27. Equations • x = x0cos(ωt) (t=0 where x = x0) • x = x0sin(ωt) (t=0 where x = 0) • v = v0cos(ωt) (t=0 where v = max) • v = v0sin(ωt) (t=0 where v = 0) • v = ω √(x02 – x2) (v = instantaneous velocity) • Lets draw a graph for each situation

28. Practice Problem 4.5 • The graph shows the variation with time t of the displacement, x of a particle undergoing simple harmonic motion. • Which graph correctly shows the variations with time, t of the acceleration a of the particle?

29. Oscillations If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.

30. Oscillations We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure). The force exerted by the spring depends on the displacement:

31. Oscillations • The minus sign on the force indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position. • k is the spring constant • The force is not constant, so the acceleration is not constant either

32. Oscillations • Displacement is measured from the equilibrium point • Amplitude is the maximum displacement • A cycle is a full to-and-fro motion; this figure shows half a cycle • Period is the time required to complete one cycle • Frequency is the number of cycles completed per second

33. Oscillations If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.

34. Simple Harmonic Motion Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator.

35. Simple Harmonic Motion 4.1.4 Solve problems using the defining equation for SHM. • Identify all the forces acting on an object when it is displaced an arbitrary distance x from its rest position using a free body diagram. • Calculate the resultant force using Newton’s second law. If this force is proportional to the displacement and always points back towards the mean position (i.e. F -x) then the motion of the object must be simple harmonic.

36. Simple Harmonic Motion 3) Once simple harmonic motion has been identified, the equation of motion must be in the following terms F=ma and F=-kx therefore ma=-kx a= -k x m Since a=-ω2x Then it can be said that ω2=k/m or ω=√(k/m)

37. The Period and Sinusoidal Nature of SHM 4.1.6 Solve problems, both graphically and by calculation, for acceleration, velocity and displacement during SHM. If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed vmax, we find that the x component of its velocity varies as: This is identical to SHM.

38. The Period and Sinusoidal Nature of SHM Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency:

39. The Period and Sinusoidal Nature of SHM We can similarly find the position as a function of time:

40. The Period and Sinusoidal Nature of SHM The top curve is a graph of the previous equation. The bottom curve is the same, but shifted ¼ period so that it is a sine function rather than a cosine.

41. The Period and Sinusoidal Nature of SHM The velocity and acceleration can be calculated as functions of time; the results are below, and are plotted at left.

42. The Simple Pendulum A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible.

43. The Simple Pendulum In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have: which is proportional to sin θ and not to θ itself. However, if the angle is small, sin θ≈ θ.

44. The Simple Pendulum where Therefore, for small angles, we have: The period and frequency are:

45. The Simple Pendulum So, as long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass.

46. Energy in the Simple Harmonic Oscillator We already know that the potential energy of a spring is given by: The total mechanical energy is then: The total mechanical energy will be conserved, as we are assuming the system is frictionless.

47. Energy in the Simple Harmonic Oscillator If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. We know what the potential energy is at the turning points: