Introduction to Simple Harmonic Motion
Introduction to Simple Harmonic Motion PHYS1033
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This 5 page Class Notes was uploaded by Oreofeoluwa on Saturday September 5, 2015. The Class Notes belongs to PHYS1033 at University of Cincinnati taught by Dr. Howard Jackson in Fall 2015. Since its upload, it has received 62 views. For similar materials see Physics of Sound and Music in Physics 2 at University of Cincinnati.
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Date Created: 09/05/15
PHY51033001 Oreofeoluwa Oyelowo PHYS 1033 Physics of Sound and Music Dr Howard Jackson Week 2 SHM Concepts of amplitude periods wavelengths simple vibrating systems resonance 12 Simple Harmonic Motion key concept in the study of waves and sound Definition SHM is a kind of periodic motion that satisfies 2 specific conditions and has a graph that s in the shape of a sine or a cosine wave Breakdown Periodic motion motion that repeats after equal time intervals The two conditions 1 An equilibrium position must be present A position at which the body remains if it is placed there and to which the body returns if it is displaced For the body to return to position E equilibrium position there must be a force F acting pulling it back This leads to condition 2 2 The force tending to pull the object back to its equilibrium position must be a linear restoring force X FOCX Where F force bringing object back to position E x distance of the object from position E and oc represents linear proportionality ie the force must get smaller as the object gets closer to its equilibrium position Sine or Cosine Wave Summary 1Page PHY51033001 Oreofeoluwa Oyelowo 1 Two conditions required to produce SHM are 1that an equilibrium position be present and 2that the force acting on the body to return it to its equilibrium position is a linear restoring force Last week we saw that the two conditions required to produce simple harmonic motion are 1 That an equilibrium position exists 2 That the force required to restore the body to its equilibrium position is a linear restoring force An illustration of a body that satisfies these conditions Example of SHM that satisfies these conditions A mass hanging on a spring Equilibrium point position of the mass when it is right below the spring but not moving at all It is said to be at rest at this point Linear restoring force lift the mass above its equilibrium position forces acting on it such as the force of the compression of the spring and the force of gravity pull it down towards equilibrium position and if the mass is pulled down below equilibrium position the spring pulls it back up towards equilibrium position If it takes 1N of force to lift the mass up by 1cm it will require 2N to lift it up 2cm 3N for 3cm and on and on Also expect the same force and distances for when the mass is pulled down from equilibrium This demonstrates the concept of a linear restoring force The force constant in this example gotten by dividing the force by the distance in cm is 1Ncm Hooke s law Amplitude A measured in units depending on the system cm for a mass on a spring pressure units like Pa for a sound wave volts for electric signal in a stereo Maximum displacement from equilibrium with no minus sign on it Insert graph to illustrate this Period T This is the time it takes for the wave to make one complete motion cycle or oscillation It is the time between the starting point and the ending point of the motion T l f where T is the period and f is the frequency It is measured in cm m etc depending on the graph ZlPage PHY51033001 Oreofeoluwa Oyelowo Frequency f This is the number of oscillations per seconds This simply means the number of time the complete wave repeats in a second f T The unit of frequency is 1s1secondor s 1 or Hertz Hz Frequency range of human hearing 20 Hz to 20 OOOHZ ZOkHz Question There was a question on frequency of oscillations in the JITT problems that came out this week This was simply asking for the frequency using the formula given above For example if the period of the motion is 105 frequency 110 01Hz f period T of the motion is 50ms you have to convert to secondsfirst 50ms 0055 f 1T 1005 20Hz Wavelength A Wavelength is the repeat distance between the start point and the end point It is measured in meters A ST where A wavelength S velocity of sound and T Period Recall T 1f Therefore A STAS1 A y f f If we multiply both sides by f so that we can isolate v on the right hand side we end up with S f A SHM amp Uniform Circular MotionO Recall that Uniform Circular Motion is an example of SHM SHM is uniform circular motion with its vertical motion projected along a line on a plane 3 2 1 PHY51033001 Oreofeoluwa Oyelowo 13 Application to Sound Higher frequencies produce higher pitches but shorter periods To make the wave sound louder increase the amplitude and decreased amplitude for the wave to sound softer Waves can be displayed using an oscilloscope This is what transmits electric signals into the wave forms that we see on the screens in class The signal from a microphone can also be turned into digital signal and made available on a screen showing its wave patterns We saw this in play when Professor Howard Jackson whistled into a microphone and we could see the wave form 14 Driven SHM and Damped SHM In SHM when a driving force is continually applied to a vibrating system causing an increase in its amplitude we arrive at a driven harmonic motion Resonance Resonance is a driven harmonic motion in which the frequency of the driving force is the same as the naturalspecialcharacteristic frequency also called f knot fo of that vibrating system Recall the fall of the Tacoma Narrows Bridge If it is driven harmonic motion applied at a frequency that is not the natural frequency it is NOT resonance Also if the amplitude starts to decrease with respect to time it is called damped harmonic motion The following experiments illustrate damped harmonic motion as well as resonance Sound and Resonance Resonance was seen in the following experiments when we were able to hear distinct pitches from the vibrating bodies 1 Metal rods of different lengths When we hit the rod at the top we could hear distinct pitches from each of those rods It sounded different for the different rods This shows that they each have their own characteristic frequency Recall that higher frequencies produce higher pitches 4Page PHY51033001 Oreofeoluwa Oyelowo Rod with blue tape Also recall that there was a rod with blue tape at the center and the sound was louder when Professor Howard Jackson held the rod at that point than when he held it closer to the top This illustrated that holding the rod at any point inhibits vibration whereas holding it at the center allows both ends to vibrate evenly which is an example of SHM 2 Tuning forks Observation 1 The tuning fork was vibrating when hit with a small hammer However after a while the sound started to die damped harmonic motion Explanation This illustrates that vibrating bodies lose energy as heat as a result of their vibration Vibrating bodies also lose energy as they move the air back and forth Observation 2 The tuning fork had a softer pitch when it was removed from the box and got louder when it was placed back in the box Explanation The frequency of the box was tuned to the natural frequency of the tuning fork Also the tuning fork and the box are both vibrating when the tuning fork comes in contact with the box Thus we hear a louder sound Question So if we have an even larger box for the tuning fork would it be louder or softer Answer Louder It is a larger object and its vibration would produce more sound Question Would the sound die away faster or more slowly Answer The sound would die away faster because it is the same frequency but a larger body is dispensing this energy more efficiently This means it would be used up more easily SIPage
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