1 Simple Harmonic Motion NCEA AS 3.4 Text: Chapters 6-8

2 SHM A periodic or repeating motion. A periodic or repeating motion. Object moves back and forth over the same path Object moves back and forth over the same path Egs pendulum, mass on spring, bobbing up and down in waves, electrons in A.C circuits, Egs pendulum, mass on spring, bobbing up and down in waves, electrons in A.C circuits,

3 Terms to know… Equilibrium position – the undisturbed position of the object (mid-point of motion) Equilibrium position – the undisturbed position of the object (mid-point of motion) Amplitude – maximum distance moved by object from equilibrium Amplitude – maximum distance moved by object from equilibrium A A EQUILIBRIUM

4 Definition of SHM What makes it different to other forms of periodic motion? What makes it different to other forms of periodic motion? There is a “restoring force” acting on the object trying to return it to it’s equilibrium positionThere is a “restoring force” acting on the object trying to return it to it’s equilibrium position The size of this restoring force (and also therefore the acceleration) is directly proportional to the displacement from equilibriumThe size of this restoring force (and also therefore the acceleration) is directly proportional to the displacement from equilibrium

5 Definition of SHM In other words… In other words…

6 SHM vs Circular Motion Another way to think of SHM is as the component in one direction of the motion of an object moving in a circle. Another way to think of SHM is as the component in one direction of the motion of an object moving in a circle. Eg compare the motion of your foot and your knee when riding a bike Eg compare the motion of your foot and your knee when riding a bike Do pg122 Qu 1-4

7 Reference Circles This is the circle of motion associated with an object doing SHM. This is the circle of motion associated with an object doing SHM. It’s centre is at equilibrium It’s centre is at equilibrium It’s radius is the amplitude of the SHM. It’s radius is the amplitude of the SHM. m p

8 Reference Circles Point p moves around the circle at constant speed and is always horizontally in line with the mass on the spring Point p moves around the circle at constant speed and is always horizontally in line with the mass on the spring The period of the SHM equals the period of the circular motion. The period of the SHM equals the period of the circular motion. p m

9 Displacement-time At any time t, displacement y can be calculated by: At any time t, displacement y can be calculated by: p m  A y

10 Displacement The displacement of an object doing SHM varies as a sine function. The displacement of an object doing SHM varies as a sine function. Eg ink pot pendulum Eg ink pot pendulum

11 Displacement-time m m m m m m m m m t =

12Displacement-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=

13 Period and frequency Just like circular motion, SHM has a frequency, period and angular frequency . Just like circular motion, SHM has a frequency, period and angular frequency .

14 Note It is really important to know where the SHM motion is at t=0s It is really important to know where the SHM motion is at t=0s If t=o at y=0 i.e. equilibrium, displacement will be a sin graph. If t=o at y=0 i.e. equilibrium, displacement will be a sin graph. If t=0 at y= + A you will have a …… displacement graph. If t=0 at y= + A you will have a …… displacement graph. What will it be for y= - A at t=0???? What will it be for y= - A at t=0???? Y=-Acosθ

15 Sam is studying his grandfathers clock. He measures the length of the pendulum 1.0m and notices it ticks at 1.0s intervals. The mass swings 3.0cm from the middle. Y= + t=0s What is the amplitude of the motion? 3.0cm What is the period of the motion? 2.0s Calculate the angular frequency

16 Sketch a displacement time graph for the motion. Label the axis t-s Y-cm

17 Write down the equation for the displacement of the motion y=0.030cos3.1t What is the displacement at t=0.25s? y=0.030cos(3.1x0.25) =2.1cm

18 Velocity For p and m to be staying in line horizontally they must both have the same vertical component to their speeds at any time. For p and m to be staying in line horizontally they must both have the same vertical component to their speeds at any time. p m  A vmvm vpvp

19 Velocity V m will be equal to the vertical component of v p. V m will be equal to the vertical component of v p. vmvm vpvp 

20 Velocity-time Graphs A phasor diagram can be used to draw velocity time graphs A phasor diagram can be used to draw velocity time graphs The velocity phasor The velocity phasor Has length =  AHas length =  A Is 90° or  /2 rads ahead of the displacement phasorIs 90° or  /2 rads ahead of the displacement phasor Displacement Phasor Velocity Phasor 

21 Velocity-time Graphs The velocity-time graph: The velocity-time graph: t ω t=6 t=7 t=0 t=1 t=3 t=4 t=

22 Velocity The maximum speed the object travels is when v m =v p =A The maximum speed the object travels is when v m =v p =A The object has this maximum speed when displacement is zero, and zero speed when displacement is maximum. The object has this maximum speed when displacement is zero, and zero speed when displacement is maximum.

23 Velocity t ω velocity displacement

24 Back to the Grandfather clock………… remember : y=0.030cos3.1t Write an equation for the velocity of the pendulum. v=-Aωsinωt v=-0.093sin3.1t What is the maximum speed? What is the speed at t=0.25s? 9.3 cms -1 v=-0.093sin(3.1x0.25) =-6.5 cms -1 What is the significance of the negative sign The pendulum is moving towards the negative side

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

26Acceleration amam apap  Negative sign indicates direction

27 Acceleration

28 Acceleration-time Graph Looks like this: Looks like this: t t=5 ω t=0 t=6 t=7 t=1 t=2 t=

29 Acceleration The acceleration phasor is 180° or  radians ahead of the displacement phasor. The acceleration phasor is 180° or  radians ahead of the displacement phasor. This is because when the displacement is at it’s greatest positive value, the acceleration is greatest back towards the centre, ie negative, and vice versa This is because when the displacement is at it’s greatest positive value, the acceleration is greatest back towards the centre, ie negative, and vice versa

30 All three…. t ω velocity displacement acceleration

31 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

32 Back to the Grandfather clock…………again remember : y=0.030cos3.1t and v=-0.093sin3.1t Now write an equation for the acceleration of the pendulum. a=-A ω 2 cos ωt a=-0.29cos3.1t What is the maximum acceleration? What is the acceleration at t=0.25s? 29 cms -2 a=-0.288cos(3.1x0.25) =-21 cms -2 What is the significance of the negative sign The pendulum is accelerating towards the negative side

33 What is the acceleration at t=0.75s? a=-0.288cos(3.1x0.75) = x = 20 cms -2 = x = 20 cms -2 Notice how the answer is now positive That means acceleration is towards the positive side

34Tides The tide rises and falls with SHM with a period of approx. 12 hours. If the difference between high and low tide is 6.0m, how long does it take to fall 1.0m from high tide? The tide rises and falls with SHM with a period of approx. 12 hours. If the difference between high and low tide is 6.0m, how long does it take to fall 1.0m from high tide?

35 1.0m 6.0m   2.0m 3.0m

36 What is the maximum speed of the tide? What is the speed of the tide when it is 1.0m out from high tide?

37 y= + A

38 Mass on Spring Where k=the spring constant measured in Nm -1 and y = the extension or compression of the spring Where k=the spring constant measured in Nm -1 and y = the extension or compression of the spring m y -y

39 Maths….

40 Maths continued… So the larger the mass, the longer the period of oscillation So the larger the mass, the longer the period of oscillation The stiffer the spring, the shorter the period of oscillation The stiffer the spring, the shorter the period of oscillation

41 Springs If more than one spring is used then k=k 1 +k 2 because the restoring force from both springs add together. If more than one spring is used then k=k 1 +k 2 because the restoring force from both springs add together. m k1k1 k2k2 m k1k1 k2k2 Do pg 134 Qu 1-4

42 Pendulums A pendulum is only SHM if  is small, so that y is approximately a straight line. A pendulum is only SHM if  is small, so that y is approximately a straight line. L L  y

43 Pendulums The forces can be resolved into components. The forces can be resolved into components. The component acting in the direction of the motion is the restoring force. The component acting in the direction of the motion is the restoring force. F resultant F tension F weight  

44 Maths See page 131 in your book to see how this formula is derived… See page 131 in your book to see how this formula is derived…

45 What length of pendulum is required to keep the same time as a clock? You want a period of one second. g=9.81ms -2 Build a pendulum and test it.

46 Pendulums Period of oscillation is independent of mass and amplitude, so pendulums are often used as timing devices. Eg grandfather clocks Period of oscillation is independent of mass and amplitude, so pendulums are often used as timing devices. Eg grandfather clocks So period of oscillation really only depends on how long the pendulum is, as g is fairly constant on earth. So period of oscillation really only depends on how long the pendulum is, as g is fairly constant on earth. Do pg 136 Qu 5-7

47 Torsional Pendulums Combination of rotational motion and SHM Combination of rotational motion and SHM proportional to  proportional to  Acceleration is given by: Acceleration is given by: Where  = 2/T Do pg 137 Qu 8,9 (hard!)

48 Energy in SHM Energy must be supplied to start SHM Energy must be supplied to start SHM During each cycle the energy changes between kinetic energy and potential energy During each cycle the energy changes between kinetic energy and potential energy

49 Energy in SHM E total E kinetic E potential t

50 Energy in SHM For all objects undergoing SHM: For all objects undergoing SHM: Potential energy is maximum and kinetic energy is zero at maximum diplacement.Potential energy is maximum and kinetic energy is zero at maximum diplacement. Potential energy is zero and kinetic energy is maximum at equilibrium.Potential energy is zero and kinetic energy is maximum at equilibrium.

51 Energy in SHM -A +A y E E total E potential E kinetic

52 Energy Reminders of energy equations: Reminders of energy equations: Do pg 146 Qu 1-5

53 Damping In practice, energy is lost due to friction during each oscillation. In practice, energy is lost due to friction during each oscillation. This reduces the total amount of energy and therefore reduces the maximum displacement that the object reaches. ie amplitude decreases. This reduces the total amount of energy and therefore reduces the maximum displacement that the object reaches. ie amplitude decreases. NOTE: Damping does NOT alter the period. NOTE: Damping does NOT alter the period.

54 Damping Max A t Decay Envelope

55 Damping Damping is measured as a percentage loss per cycle. Damping is measured as a percentage loss per cycle. Eg. An object with amplitude 1m and 20% damping: Eg. An object with amplitude 1m and 20% damping: 1m, 0.8m, 0.64m, 0.512m, 0.41m (Each value is 80% of previous one) 1m, 0.8m, 0.64m, 0.512m, 0.41m (Each value is 80% of previous one)

56 Damping Damping can be heavy or light depending on the situation: Damping can be heavy or light depending on the situation: Eg car suspension, balance arms, clock pendulums,Eg car suspension, balance arms, clock pendulums, t t

57 Forced Oscillations If an object is free to move, a driving force can be applied to make it oscillate. If an object is free to move, a driving force can be applied to make it oscillate. The frequency of the oscillation equals the frequency of the driving force. The frequency of the oscillation equals the frequency of the driving force. The amplitude can vary, depending on what frequency is driving it. The amplitude can vary, depending on what frequency is driving it.

58 Forced Oscillations A Natural frequency Driving Frequency

59 Resonance If the driving frequency equals the objects natural frequency, the amplitude gets very large. If the driving frequency equals the objects natural frequency, the amplitude gets very large. This is called Resonance This is called Resonance Resonance can be useful (eg musical instruments, tuning radio circuits) or problematic (eg building or bridge resonance in winds or earthquakes) Resonance can be useful (eg musical instruments, tuning radio circuits) or problematic (eg building or bridge resonance in winds or earthquakes) Do pg 147 Qu 6-10

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