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- 1. Waves, Optics, Oscillation, and Gravitation By: Charnae’ Kearney and Andy Hurst
- 2. Traveling Wave <ul><li>Any kind of wave which propagates in a single direction with negligible change in shape. </li></ul><ul><li>Traveling waves are observed when a wave is not confined to a given space along the medium. The most commonly observed traveling wave is an ocean wave </li></ul>
- 3. Traveling and Standing Waves <ul><li>An important class of traveling waves is plane waves in air which create standing waves in rectangular enclosures such as ``shoebox'' shaped concert halls. </li></ul><ul><li>Standing waves don't go anywhere, but they do have regions where the disturbance of the wave is quite small, almost zero. These locations are called nodes. There are also regions where the disturbance is quite intense, greater than anywhere else in the medium, called antinodes. </li></ul>
- 4. Wave Propagation <ul><li>Any of the waves that waves travel </li></ul><ul><li>With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves. </li></ul><ul><li>For electromagnetic waves, propagation may occur in a vacuum as well as in a material medium. Most other wave types cannot propagate through vacuum and need a transmission medium to exist. </li></ul><ul><li>Another useful parameter for describing the propagation is the wave velocity that mostly depends on some kind of density of the medium. </li></ul>
- 5. Principle of Superposition <ul><li>The regions where they overlap, the resultant displacement is the algebraic sum of their separate displacements. </li></ul>
- 6. Simple Harmonic Motion <ul><li>Regular, repeated, friction-free motion in which the restoring force has the mathematical form F= -kx </li></ul><ul><li>Common examples: mass on a spring and a pendulum </li></ul><ul><li>The word “harmonic” refers to the motion being sinusoidal, it is “simple” when there is pure sinusoidal motion of a single frequency </li></ul><ul><li>As an object vibrates in harmonic motion, energy is transferred between potential and kinetic energy. </li></ul>
- 7. Mass on a Spring <ul><li>When it vibrates it has both a period and a frequency </li></ul><ul><li>Restoring force – the force trying to restore it (mass on a spring) back towards the center of the oscillation </li></ul>
- 8. Pendulum <ul><li>A mass on the end of a string which oscillates in harmonic motion </li></ul><ul><li>T= 2 π√ L/G </li></ul><ul><li>L is the length of the pendulum </li></ul><ul><li>G is the acceleration due to gravity </li></ul>
- 9. Newton’s Law of Gravity <ul><li>Every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. </li></ul><ul><li>F= G m 1 m 2 </li></ul><ul><li>------------------------- </li></ul><ul><li>r ² </li></ul> , F is the force between the masses, G is the gravitational constant, m 1 is the first mass, m 2 is the second mass, and r is the distance between the masses.
- 10. Newton’s Law of Gravity Contin. <ul><li>Gravitation is a UNIVERSAL force between all objects in the universe. </li></ul>
- 11. Circular Orbits of Planets & Satellites <ul><li>As a satellite orbits the earth, it is pulled toward the earth with a gravitational force which is acting as a centripetal force. The inertia of the satellite causes it to tend to follow a straight-line path, but the centripetal gravitational force pulls it toward the center of the orbit. </li></ul><ul><li>If a satellite of mass m moves in a circular orbit around a planet of mass M , we can set the centripetal force equal to the gravitational force and solve for the speed of the satellite orbiting at a particular distance r : </li></ul>
- 12. General Orbits of Planets & Satellites <ul><li>Elliptical Motion: </li></ul><ul><li>Kepler’s Law of Planetary Motion </li></ul><ul><li>The orbit of every planet is an ellipse with the Sun at one of the two foci. </li></ul><ul><li>A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. [1] </li></ul><ul><li>The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit </li></ul>
- 13. Important Key Terms to Remember <ul><li>Period - (T) of the motion is the time required for the motion to repeat. </li></ul><ul><li>Frequency - (f) refers to the number of complete repetitions of the motion that occur each second. The frequency is inversely related to the period. </li></ul><ul><li>Simple harmonic motion - (SHM) refers to periodic vibrations or oscillations that exhibit two characteristics: 1) the force acting on the object and the magnitude of the object’s acceleration are always directly proportional to the displacement of the object from its equilibrium position, and 2) both the force vector and the acceleration vector are directed opposite to the displacement vector and therefore in toward the object’s equilibrium position. </li></ul>
- 14. Important Key Terms to Remember Cont. <ul><li>Simple pendulum - is assumed to have its entire mass concentrated at the end of its length. The simple pendulum undergoes SHM if the maximum angle that it is displaced from equilibrium is small (approximately 15 ° or less). </li></ul><ul><li>Principle of superposition - states that when two waves pass through a medium at the same time, the resultant displacement of the medium at any particular moment of time equals the algebraic sum of the displacement of the component waves at that point. </li></ul><ul><li>Standing waves - are produced by the superposition of two periodic waves having identical frequencies and amplitudes which are traveling in opposite directions. </li></ul>
- 15. Important Formulas <ul><li>K=(F/x) </li></ul><ul><li>f=(1/T) </li></ul><ul><li>V=f </li></ul><ul><li>F=-kx </li></ul><ul><li>T=2 π (m/k) </li></ul><ul><li>T=2 π (L/g) </li></ul><ul><li>V= λ f (wave velocity is = to the product of wavelength and frequency) </li></ul>
- 16. Example #1 <ul><li>A spring of constant k = 100 N/m hangs at its natural length from a fixed stand. A mass of 3 kg is hung on the end of the spring, and slowly let down until the spring and mass hang at their new equilibrium position. x </li></ul><ul><li>(a) Find the value of the quantity x in the figure above. </li></ul><ul><li>The spring is now pulled down an additional distance x and released from rest. </li></ul><ul><li>(b) What is the potential energy in the spring at this distance? </li></ul><ul><li>(c) What is the speed of the mass as it passes the equilibrium position? </li></ul><ul><li>(d) How high above the point of release will the mass rise? </li></ul><ul><li>(e) What is the period of oscillation for the mass? </li></ul>x x
- 17. Example #1 Solution <ul><li>(a) As it hangs in equilibrium, the upward spring force must be equal and opposite to the downward weight of the block. </li></ul><ul><li>(b) The potential energy in the spring is related to the displacement from equilibrium position by the equation </li></ul><ul><li>(c) Since energy is conserved during the oscillation of the mass, the kinetic energy of the mass as it passes through the equilibrium position is equal to the potential energy at the amplitude. Thus, </li></ul><ul><li>(d) Since the amplitude of the oscillation is 0.3 m, it will rise to 0.3 m above the equilibrium position. </li></ul><ul><li>(e) </li></ul>F s m g
- 18. Example #2 <ul><li>A string is attached to a vibrating machine which has a frequency of 120 Hz. The other end of the string is passed over a pulley of negligible mass and friction and is attached to a weight hanger which holds a mass m = 0.5 kg. </li></ul><ul><li>(a) Determine the tension in the string. </li></ul><ul><li>(b) The speed of the wave in the string is related to the tension by the equation </li></ul><ul><li>, where F T is the tension in the string and μ is the linear density of the string. If the linear density of this string is 0.05 kg/m, determine the speed of the wave in the string. </li></ul><ul><li>(c) Determine the wavelength of the wave in the string. </li></ul><ul><li>(d) Determine the length of the string from the point of attachment on the vibrating machine to the pulley. </li></ul><ul><li>(e) Would you need to increase or decrease the mass on the hanger to produce a lower number of loops? Explain. </li></ul>m L m L
- 19. Example #2 Solution <ul><li>(a) </li></ul><ul><li>(b) </li></ul><ul><li>(c) </li></ul><ul><li>(d) </li></ul><ul><li>(e) A lower number of loops would imply a longer wavelength, which would require a higher speed, which would require a higher tension in the string, which would require increasing the mass on the hanger. </li></ul>
- 20. Example #3 <ul><li>A pendulum of mass 0.4 kg and length 0.6 m is pulled back and released from and angle of 10˚ to the vertical. </li></ul><ul><li>(a) What is the potential energy of the mass at the instant it is released. Choose potential energy to be zero at the bottom of the swing. </li></ul><ul><li>(b) What is the speed of the mass as it passes its lowest point? </li></ul><ul><li> </li></ul><ul><li>This same pendulum is taken to another planet where its period is 1.0 second. </li></ul><ul><li>(c) What is the acceleration due to gravity on this planet? </li></ul>
- 21. Example #3 Solution <ul><li>(a) First we must find the height above the lowest point in the swing at the instant the pendulum is released. </li></ul><ul><li> Recall from chapter 1 of this study guide </li></ul><ul><li>that </li></ul><ul><li>Then </li></ul><ul><li> </li></ul><ul><li>(b) Conservation of energy: </li></ul><ul><li>(c) </li></ul>L h 10˚ L

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