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1 Simple Harmonic Motion NCEA AS 3.4 Text: Chapters 6-8 Slide 2 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, Slide 3 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 Slide 4 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 its equilibrium positionThere is a restoring force acting on the object trying to return it to its 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 Slide 5 5 Definition of SHM In other words In other words Slide 6 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 Slide 7 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. Its centre is at equilibrium Its centre is at equilibrium Its radius is the amplitude of the SHM. Its radius is the amplitude of the SHM. m p Slide 8 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 Slide 9 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 Slide 10 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 Slide 11 11 Displacement-time m m m m m m m m m t = 0 1 2 3 4 5 6 7 8 Slide 12 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 t=0 t=1 t=2 t=3 t=5 t=6 t=7 17235468 Slide 13 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 . Slide 14 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=-Acos Slide 15 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= + A @ t=0s What is the amplitude of the motion? 3.0cm What is the period of the motion? 2.0s Calculate the angular frequency Slide 16 16 Sketch a displacement time graph for the motion. Label the axis t-s 0.51.01.52.0 Y-cm -3.0 3.0 Slide 17 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 Slide 18 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 Slide 19 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 Slide 20 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 Slide 21 21 Velocity-time Graphs The velocity-time graph: The velocity-time graph: t t=6 t=7 t=0 t=1 t=3 t=4 t=5 17235468 Slide 22 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. Slide 23 23 Velocity t 17235468 velocity displacement Slide 24 24 Back to the Grandfather clock remember : y=0.030cos3.1t Write an equation for the velocity of the pendulum. v=-Asint 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 Slide 25 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 Slide 26 26Acceleration amam apap Negative sign indicates direction Slide 27 27 Acceleration Slide 28 28 Acceleration-time Graph Looks like this: Looks like this: t t=5 t=0 t=6 t=7 t=1 t=2 t=3 17235468 Slide 29 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 its greatest positive value, the acceleration is greatest back towards the centre, ie negative, and vice versa This is because when the displacement is at its greatest positive value, the acceleration is greatest back towards the centre, ie negative, and vice versa Slide 30 30 All three. t 17235468 velocity displacement acceleration Slide 31 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 Slide 32 32 Back to the Grandfather clockagain remember : y=0.030cos3.1t and v=-0.093sin3.1t Now write an equation for the acceleration of the pendulum. a=-A 2 cos t 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 Slide 33 33 What is the acceleration at t=0.75s? a=-0.288cos(3.1x0.75) =-0.288 x -0.6847 = 20 cms -2 =-0.288 x -0.6847 = 20 cms -2 Notice how the answer is now positive That means acceleration is towards the positive side Slide 34 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? Slide 35 35 1.0m 6.0m 2.0m 3.0m Slide 36 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? Slide 37 37 t=0 @ y= + A Slide 38 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 Slide 39 39 Maths. Slide 40 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 Slide 41 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 Slide 42 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 Slide 43 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 Slide 44 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 Slide 45 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. Slide 46 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 Slide 47 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!) Slide 48 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 Slide 49 49 Energy in SHM E total E kinetic E potential t Slide 50 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. Slide 51 51 Energy in SHM -A +A y E E total E potential E kinetic Slide 52 52 Energy Reminders of energy equations: Reminders of energy equations: Do pg 146 Qu 1-5 Slide 53 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. Slide 54 54 Damping Max A t Decay Envelope Slide 55 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) Slide 56 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 Slide 57 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. Slide 58 58 Forced Oscillations A Natural frequency Driving Frequency Slide 59 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 Slide 60 60