Simple Harmonic & Circular Motion


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Simple Harmonic & Circular Motion

  1. 1. Oscillations and Waves Topic 4.1 Kinematics of simple harmonic motion (SHM)
  2. 2. Examples of oscillations <ul><li>A periodic motion is one during which a body continually retraces its path at equal intervals </li></ul>Nature of oscillating system p.e. stored as k.e. possessed by moving Mass on helical spring Elastic energy of spring Mass cantilever Elastic energy of bent rod Rod Simple pendulum Gravitational p.e. of bob Bob Vertical rod floating in liquid of zero viscosity Gravitational p.e. of rod or liquid rod
  3. 3. Displacement <ul><li>Its position x as a function of time t is: where A is the amplitude of motion: the distance from the centre of motion to either extreme </li></ul><ul><li>Its position x as a function of time t </li></ul>O B A P
  4. 4. Amplitude & Period <ul><li>Which ball has a larger amplitude? </li></ul>Ball A <ul><li>Which ball has the larger period? </li></ul>Ball A <ul><li>T is the period of motion: the time for one complete cycle of the motion </li></ul>
  5. 5. Frequency <ul><li>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: </li></ul><ul><li>If ball B has a time period of 12 s, what is the frequency? </li></ul>f = 0.0833 Hz
  6. 6. Angular frequency <ul><li>Angular frequency is the rotational analogy to frequency. Represented as  , 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 </li></ul><ul><li>Which ball has the larger angular frequency? </li></ul>Ball B
  7. 7. <ul><li>Displayed below is a position-time graph of a piston moving in and out. </li></ul>Find the: Amplitude Period Frequency Angular frequency 10 cm 0.2 s 5.0 Hz 10  rads -1
  8. 8. Phase <ul><li>Here is an oscillating ball. </li></ul><ul><li>Its motion can be described as follows: </li></ul><ul><ul><li>Then it moves with v < 0 through the centre to the left </li></ul></ul><ul><ul><li>Then it is at v = 0 at the left </li></ul></ul><ul><ul><li>Then it moves with v > 0 through the centre to the right </li></ul></ul><ul><ul><li>Then it repeats... </li></ul></ul>
  9. 9. Phase This information concentrates on what phase of the cycle is being executed. It is not concerned with the particulars of amplitude. Mathematically, the phase is the &quot;w t&quot; in: x(t) = A cos (  t)
  10. 10. Phase <ul><li>Here is an oscillating ball. </li></ul>Recall: x(t) = A cos (  t)  What phase is the ball in when: x = 0, v < 0 A. 0.00  rad B. 0.25  rad C. 0.50  rad D. 1.0  rad E. 1.5  rad F. 1.7  rad
  11. 11. Phase <ul><li>Here is an oscillating ball. </li></ul>Recall: x(t) = A cos (  t)  What phase is the ball in when: x = +A, v = 0 A. 0.00  rad B. 0.25  rad C. 0.50  rad D. 1.0  rad E. 1.5  rad F. 1.7  rad
  12. 12. SHM and circular motion Uniform Circular Motion (radius A, angular velocity w) Simple Harmonic Motion (amplitude A, angular frequency w) <ul><li>Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis </li></ul><ul><li>The phase angle  t in SHM corresponds to the real angle  t through which the ball has moved in circular motion. </li></ul>
  13. 13. Velocity and acceleration <ul><li>Once you know the position of the oscillator for all times, you can work out the velocity and acceleration functions. </li></ul>x(t) = A cos (  t +  ) <ul><li>The velocity is the time derivative of the position so: </li></ul>v(t) = -A  sin (  t +  ) <ul><li>The Change from cos to sin means that the velocity is 90 o out of phase with the displacement </li></ul><ul><li>when x = 0 the velocity is a maximum and when x is a minimum v = 0 </li></ul><ul><li>The acceleration is the time derivative of the velocity so: </li></ul>a(t) = -A  2 cos (  t +  ) <ul><li>Notice also from the preceding that: a(t) = -  2 x   </li></ul><ul><li>The acceleration is exactly out of phase with the displacement. </li></ul>
  14. 14. Velocity and acceleration <ul><li>Watch the oscillating duck. Let's consider velocity now </li></ul><ul><li>Remember that velocity is a vector, and so has both negative and positive values. </li></ul><ul><li>Where does the magnitude of v(t) have a maximum value? </li></ul><ul><li>Where does v(t) = 0? </li></ul>C A and E
  15. 15. Velocity and acceleration <ul><li>Watch the oscillating duck. Let's consider acceleration now </li></ul><ul><li>Remember that acceleration is a vector, and so has both negative and positive values. </li></ul><ul><li>Where does the magnitude of a(t) have a maximum value? </li></ul><ul><li>Where does a(t) = 0? </li></ul>A and E C
  16. 16. Summary 1 <ul><li>You now know several parameters that are used to describe SHM: </li></ul><ul><ul><ul><ul><li>amplitude (A) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>period (T) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>frequency (f) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>angular frequency (  ) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>initial phase (  ) </li></ul></ul></ul></ul><ul><ul><ul><ul><li>maximum velocity (v(t)max) and </li></ul></ul></ul></ul><ul><ul><ul><ul><li>maximum acceleration (a(t)max) </li></ul></ul></ul></ul>
  17. 17. Questions <ul><li>1. A pendulum has maximum horizontal displacement of 0.10m. It oscillates with a period of 2s. </li></ul><ul><li>What is it’s displacement at time 0.5s </li></ul><ul><li>What is it’s displacement at time 1.3s </li></ul><ul><li>What is the maximum velocity </li></ul><ul><li>What is the maximum acceleration </li></ul><ul><li>2. A surfer bobs up and down on the surface of a wave with a period of 4.0s and an amplitude of 1.5m. </li></ul><ul><li>What is the surfer’s maximum acceleration </li></ul><ul><li>What is the surfer’s maximum velocity </li></ul>
  18. 18. Questions <ul><li>3. An object moving with SHM has an amplitude of 2cm and a frequency of 20 Hz </li></ul><ul><ul><li>What is it’s period </li></ul></ul><ul><ul><li>What is its acceleration at the middle and end of an oscillation </li></ul></ul><ul><ul><li>What are the velocities at the middle and end </li></ul></ul><ul><li>4. A steel strip clamped at one end, oscillates with frequency of 50Hz and has an amplitude of 8mm </li></ul><ul><ul><li>What is its period </li></ul></ul><ul><ul><li>What is its angular frequency </li></ul></ul><ul><ul><li>What is its velocity at the middle and end of an oscillation </li></ul></ul><ul><ul><li>What are the corresponding accelerations </li></ul></ul>
  19. 19. Summary 2 <ul><li>There are many relations among these parameters. One minimum set of parameters to completely specify the motion is: </li></ul><ul><ul><ul><ul><ul><li>amplitude (A) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>angular frequency (  ) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>initial phase (  ) </li></ul></ul></ul></ul></ul><ul><li>You are already familiar with this set, which is used in: </li></ul><ul><li>x(t) = A cos (  t +  ) </li></ul><ul><li>The trick in solving SHM problems is to take the given information, and use it to extract A,  and  . Once you have A,  and  , you can calculate anything. </li></ul>
  20. 20. Summary 3 <ul><li>For a given body with SHM </li></ul>In terms of time In terms of displacement Displacement Velocity Acceleration
  21. 21. <ul><li>There are 2 practical examples of SHM </li></ul><ul><li>The simple pendulum </li></ul><ul><li>Mass on a vertical spring </li></ul><ul><li>Mass on vertical spring </li></ul><ul><li>Uses Hooke’s Law F = kx </li></ul><ul><li>And F=ma </li></ul><ul><li>To give T = 2 π√ (m/k) </li></ul>
  22. 22. Simple pendulum <ul><li>A pendulum exhibits SHM </li></ul><ul><li>There is a component of the weight of the bob acting towards the centre of the motion </li></ul><ul><li>We can use Newton's 1 st and 2 nd law to help us </li></ul><ul><li>T + Wcos θ =0 perpendicular </li></ul><ul><li>and </li></ul><ul><li>Wsin θ = -ma horizontally </li></ul>W T θ
  23. 23. <ul><li>But W = mg </li></ul><ul><li>So, </li></ul><ul><li>mg sin θ = -ma </li></ul><ul><li>g sin θ = -a </li></ul><ul><li>If sin θ = θ then, θ = x/l </li></ul><ul><li>g x / l = -a </li></ul><ul><li>But we know for SHM </li></ul><ul><li>a = - w 2 x </li></ul><ul><li>So, w 2 = g/l </li></ul><ul><li>Giving, w = √(g/l) </li></ul><ul><li>More usually written in relation to period as </li></ul><ul><ul><ul><li>T = 2 π√ (l/g) </li></ul></ul></ul>
  24. 24. <ul><li>We can use this equation to establish a value for g </li></ul><ul><li>Practical: </li></ul><ul><li>Set up simple pendulum </li></ul><ul><li>Decide on a set point in the oscillation for counting </li></ul><ul><li>Measure the time taken for about 20 oscillations for 5 lengths upto 50cm </li></ul><ul><li>Calculate period </li></ul><ul><li>Plot length against period squared </li></ul><ul><li>Use gradient to find g….gradient = 4 π 2 /g </li></ul>
  25. 25. Oscillations and Waves Topic 4.2 Energy changes during SHM
  26. 26. <ul><li>The frequency of simple harmonic motion like a mass on a spring is determined by the mass m and the stiffness of the spring expressed in terms of a spring constant k ( see Hooke’s Law): </li></ul>
  27. 27. Mass on spring resonance <ul><li>A mass on a spring has a single resonant frequency determined by its spring constant k and the mass m. Using Hooke’s law and neglecting damping and the mass of the spring, Newton’s second law gives the equation of motion: </li></ul>the expression for the resonant vibrational frequency: This kind of motion is called simple harmonic motion and the system a simple harmonic oscillator.
  28. 28. Mass on spring: motion sequence <ul><li>A mass on a spring will trace out a sinusoidal pattern as a function of time, as will any object vibrating in simple harmonic motion. One way to visualize this pattern is to walk in a straight line at constant speed while carrying the vibrating mass. Then the mass will trace out a sinusoidal path in space as well as time. </li></ul>
  29. 29. Energy in mass on spring <ul><li>The simple harmonic motion of a mass on a spring is an example of an energy transformation between potential energy and kinetic energy. In the example below, it is assumed that 2 joules of work has been done to set the mass in motion. </li></ul>
  30. 30. Potential energy At extension x:
  31. 31. Kinetic energy At extension x:
  32. 32. Total energy Total energy
  33. 33. Oscillations and Waves Topic 4.3 Forced oscillations and resonance
  34. 34. Damped oscillations <ul><li>When a system executes true SHM then </li></ul><ul><ul><li>Its period is independent of its amplitude </li></ul></ul><ul><ul><li>Its total energy remains constant in time </li></ul></ul><ul><li>In practice many bodies execute approximate SHM because </li></ul><ul><ul><li>F is not proportional to x </li></ul></ul><ul><ul><li>The energy of the system decreases in time </li></ul></ul>Total energy E / J Time / s Energy of system Total energy E / J Time / s Dissipated energy
  35. 35. Damping <ul><li>Damping is caused by dissipative forces, such as air viscosity, and work is taken from the energy of oscillation. </li></ul><ul><li>Damping is the process whereby energy is taken from the oscillating system </li></ul><ul><li>For example a playground swing </li></ul><ul><ul><li>If you push it will oscillate </li></ul></ul><ul><ul><li>It will eventually slow down as energy is lost to friction </li></ul></ul><ul><ul><li>Energy needs to be supplied to keep it oscillating, that comes from you! </li></ul></ul>
  36. 36. Types of damping 1 <ul><li>Slight damping </li></ul><ul><li>This results in a definite oscillation, but the amplitude decays exponentially </li></ul>t/s x/m
  37. 37. Types of damping 2 <ul><li>Critical damping </li></ul><ul><li>No real oscillation, the time taken for displacement to become zero is a minimum </li></ul>t/s x/m T/4
  38. 38. Types of damping 3 <ul><li>Heavy damping </li></ul><ul><li>Damping force is much greater than the critical damping. The system returns to zero very slowly </li></ul>t/s x/m Very slow return to zero displacement
  39. 39. Natural oscillations <ul><li>The oscillations so far have been free oscillations, or natural oscillations, which the system has been given some energy and left alone. </li></ul><ul><li>The frequency of oscillation depends on the inertia and elasticity factors of the system </li></ul><ul><li>For example </li></ul><ul><ul><li>Guitar string, it will always play the same notes regardless of how hard you pluck it </li></ul></ul><ul><ul><li>A child’s swing, it will always swing at the same rate regardless of how faster you push it </li></ul></ul><ul><li>This is called the natural frequency, f 0 </li></ul>
  40. 40. Forced oscillations <ul><li>Previously the oscillations have been given a single push to start them moving </li></ul><ul><li>Often oscillations are subjected to a constant force, called the driving force, f </li></ul><ul><li>The effect that the driving for has depends on its frequency </li></ul>
  41. 41. Effects of forced oscillations <ul><li>The damping of the system has these effects: </li></ul><ul><li>Amplitude </li></ul><ul><ul><li>The amplitude is decreases with damping (cuts down the sharp peak) </li></ul></ul><ul><ul><li>The maximum amplitude is at a frequency less than the natural frequency </li></ul></ul><ul><li>Energy </li></ul><ul><ul><li>The power of the driver is controlled by damping </li></ul></ul>
  42. 42. Resonance <ul><li>Resonance occurs when the an oscillator is acted upon by a driving force that has the same frequency as the natural frequency </li></ul><ul><li>The driving force easily transfers its energy to the oscillator </li></ul><ul><li>From the picture the amplitude of oscillation will become very high </li></ul><ul><li>This can be a useful and sometimes very bad </li></ul>
  43. 43. Useful resonance <ul><li>Electricity, tuning a radio </li></ul><ul><ul><li>The natural frequency of the radio circuit is made equal to the incoming electromagnetic wave by changing its capacitance </li></ul></ul><ul><ul><li>The electrons in the circuit will oscillate with the incoming electromagnetic wave. </li></ul></ul><ul><ul><li>The electric current will oscillate and this can be turned into sound, through a speaker </li></ul></ul><ul><li>Microwave ovens </li></ul><ul><ul><li>Microwaves are produced at the same frequency as the natural frequency of water molecules </li></ul></ul><ul><ul><li>Water molecules absorb the energy from the microwaves and transfer their energy to the food in the form of thermal energy </li></ul></ul>
  44. 44. When resonance goes bad <ul><li>A Driving force at resonance increases the oscillations, sometimes this is unwanted </li></ul><ul><li>Structures </li></ul><ul><ul><li>Tacoma Narrows bridge, this bridge was destroyed as the wind (driving force) was at the same as the natural frequency. The bridge vibrated and shook itself apart </li></ul></ul><ul><ul><li>Tower blocks, the same effect as the bridge the wind, or earthquakes, can cause vibrations to destroy the buildings </li></ul></ul><ul><ul><li>This can be stopped by designing the building with heavy damping </li></ul></ul><ul><ul><li>High stiffness </li></ul></ul><ul><ul><li>Large mass </li></ul></ul><ul><ul><li>Shape </li></ul></ul><ul><ul><li>Good at absorbing energy </li></ul></ul>
  45. 45. Tacoma Bridge
  46. 46. Movie
  47. 47. Circular Motion <ul><li>A body moving with uniform speed in a circle is changing velocity as the direction changes. </li></ul><ul><li>This change of velocity, and the acceleration is directed towards the centre of the circle </li></ul><ul><li>This acceleration is called the centripetal acceleration </li></ul>
  48. 48. Vector Diagrams uniform speed velocity at a tangent Acceleration and force directed towards the centre
  49. 49. Centripetal Acceleration <ul><li>The expression for centripetal acceleration is </li></ul><ul><li>a = v 2 = 4  2 r r T 2 </li></ul><ul><ul><li>Where v is the velocity at any instant (i.e.the constant speed) </li></ul></ul><ul><ul><li>And r is the radius of the circle </li></ul></ul><ul><ul><li>T is the time period of one complete circle </li></ul></ul>
  50. 50. Centripetal Force <ul><li>The expression for centripetal force is F = mv 2 r </li></ul><ul><ul><li>Where v is the velocity at any instant (i.e.the constant speed) </li></ul></ul><ul><ul><li>m is the mass of the object </li></ul></ul><ul><ul><li>And r is the radius of the circle </li></ul></ul>
  51. 51. <ul><li>Examples of Centripetal Force include </li></ul><ul><ul><li>the gravitational force keeping the moon in its orbit </li></ul></ul><ul><ul><li>the friction acting sideways on the tyres of a car turning a corner </li></ul></ul><ul><ul><li>the tension in a rope, when a bucket of water is swung around your head </li></ul></ul>
  52. 52. SHM <ul><li>Useful references: </li></ul><ul><li> </li></ul><ul><li> </li></ul><ul><li> </li></ul>