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Transcript
Simple Harmonic Motion First things first… Permission slips for Weds? I need them. We’ll meet out front at the start of the day. Don’t go to your 1st period class. Choose one aspect of The Martian that relates to physics and explain the physics behind it. Typed – due 10/14. At least one page. Simple Harmonic Motion Like all motion, it’s all about displacement, velocity and acceleration Simple Harmonic Motion Like all motion, it’s all about displacement, velocity and acceleration I don’t care! Get me out of here! Topic 4: Waves – Simple Harmonic Motion 4.1 – Oscillations Oscillations Oscillations are vibrations which repeat themselves. v=0 v = 0 EXAMPLE: Oscillations v=0 v = vmax v=0 EXAMPLE: Oscillations can be driven internally, like a mass on a spring. FYI In all oscillations, v = 0 at the extremes… and v = vmax in the middle of the motion. v = vmax can be driven externally, like a pendulum in a gravitational field. x Topic 4: Waves 4.1 – Oscillations Oscillations Oscillations are vibrations which repeat themselves. EXAMPLE: Oscillations can be very rapid vibrations such as in a plucked guitar string or a tuning fork. Topic 4: Waves 4.1 – Oscillations Time period, amplitude and displacement Consider a mass on a spring that is displaced 4 meters to the right x0 and then released. We call the maximum displacement x0 the amplitude. In this example x0 = 4 m. We call the point of zero displacement the equilibrium position. Displacement x is measured from equilibrium. The period T (measured in s) is the time it takes for the mass to make one full oscillation or cycle. For this particular oscillation, the period T is about 24 seconds (per cycle). x Topic 4: Waves 4.1 – Oscillations Time period and frequency The frequency f (measured in Hz or cycles / s) is defined as how many cycles (oscillations, repetitions) occur each second. Since period T is seconds per cycle, frequency must be 1 / T. f=1/T T=1/f relation between T and f EXAMPLE: The cycle of the previous example repeated each 24 s. What are the period and the frequency of the oscillation? SOLUTION: The period is T = 24 s. The frequency is f = 1 / T = 1 / 24 = 0.042 Hz The period of a pendulum: Always remember: Do you see mass in there? What does it depend on? increase L = bigger T decrease g = bigger T Topic 4: Waves 4.1 – Oscillations Phase difference PRACTICE: Two identical mass-spring systems are started in two different ways. What is their phase difference? Start stretched and then release x Start unstretched with a push left x SOLUTION: The phase difference is one-quarter of a cycle. Topic 4: Waves 4.1 – Oscillations Phase difference PRACTICE: Two identical mass-spring systems are started in two different ways. What is their phase difference? Start stretched and then release x Start unstretched with a push right x SOLUTION: The phase difference is three-quarters of a cycle. Simple Harmonic Motion: 1. The object moves back and forth around a reference point, or mean position. 2. A restoring force acts on the object. 3. At any given point, its acceleration is directly proportional to its displacement in magnitude, but is in the opposite direction and always directed to the mean position. Topic 4: Waves 4.1 – Oscillations Conditions for simple harmonic motion In simple harmonic motion (SHM), a and x are related in a very precise way: Namely, a -x. a -x definition of SHM PRACTICE: Show that a mass oscillating on a spring executes simple harmonic motion. x As the object moves with –x, what is the acceleration’s sign? Remember – the object is slowing and stopping at xmax. F Topic 4: Waves 4.1 – Oscillations x x 0 Conditions for simple harmonic motion a -x F F and x oppose each other. definition of SHM The minus sign in Hooke’s law, F = -kx, tells us that if the displacement x is positive (right), the spring force F is negative (left). It also tells us that if the displacement x is negative (left), the spring force F is positive (right). Any force that is proportional to the opposite of a displacement is called a restoring force. For any restoring force F -x. Since F = ma we see that ma -x, or a -x. All restoring forces can drive simple harmonic motion (SHM). x Like this… Try one. • Change the length of the arm of the pendulum. • Measure the period • What kind of relationship do we see? Topic 4: Waves 4.1 – Oscillations Conditions for simple harmonic motion If we place a pen on the oscillating mass, and pull a piece of paper at a constant speed past the pen, we trace out the displacement vs. time graph of SHM. x SHM traces out perfect sinusoidal waveforms. t Note that the period can be found from the graph: Just look for repeating cycles. Topic 4: Waves 4.1 – Oscillations Qualitatively describing the energy changes taking place during one cycle of an oscillation Consider the pendulum to the right which is placed in position and held there. Let the green rectangle represent the potential energy of the system. Let the red rectangle represent the kinetic energy of the system. Because there is no motion yet, there is no kinetic energy. But if we release it, the kinetic energy will grow as the potential energy diminishes. A continuous exchange between EK and EP occurs. Topic 4: Waves 4.1 – Oscillations Qualitatively describing the energy changes taking place during one cycle of an oscillation Consider the mass-spring system shown here. The mass is pulled to the right and held in place. Let the green rectangle represent the potential energy of the system. Let the red rectangle FYI If friction and drag are represent the kinetic energy of the system. both zero ET = CONST. A continuous exchange between EK and EP occurs. Note that the sum of EK and EP is constant. EK + EP = ET = CONST relation between EK and EP x Topic 4: Waves 4.1 – Oscillations Qualitatively describing the energy changes taking place during one cycle of an oscillation EK + EP = ET = CONST relation between EK and EP Energy If we plot both kinetic energy and potential energy vs. time for either system we would get the following graph: time x v=0 v = vMAX v=0 Topic 4: Waves 4.1 – Oscillations x -2.0 0.0 2.0 Sketching and interpreting graphs of simple harmonic motion examples EXAMPLE: The displacement x vs. time t for a system undergoing SHM is shown here. x-black (+) ( -) (+) ( -) (+) v-red (different scale) t Sketch in red the velocity vs. time graph. SOLUTION: At the extremes, v = 0. At x = 0, v = vMAX. The slope determines sign of vMAX. Or – in a pendulum v=0 v = vMAX v=0 Topic 4: Waves 4.1 – Oscillations x -2.0 0.0 2.0 Sketching and interpreting graphs of simple harmonic motion examples EXAMPLE: The displacement x vs. time t for a system undergoing SHM is shown here. x-black v-red (different scale) t a-blue (different scale) Sketch in blue the acceleration vs. time graph. SOLUTION: Since a -x, a is just a reflection of x. Note: x is a sine, v is a cosine, and a is a – sine wave. SHM and Circles… SHM is a projection of circular motion Examples… For more: • The Physics Classroom (online) – Waves: Lesson 0 – Vibrations • Homer – pg 115 – 123 • For next time: Homer: 44 – 61 & homework sheet.