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harmonicwaves

# harmonicwaves - Physics 153 2006 Week 8 Simple Harmonic...

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Unformatted text preview: Physics 153 2006 Week 8: Simple Harmonic Motion • Period, Amplitude and Frequency • Simple Harmonic Motion • Vertical Springs • Total Energy, Potential Energy and Work Energy Theorem Physics 153 2006 Periodic Motion Periodic Motion - motion that repeats itself cyclically. Period - T - the time over which the motion repeats, or the time of one cycle. Frequency - f- number of cycles per unit time; f = 1/T Amplitude - A- the maximum displacement from equilibrium. Physics 153 2006 Example: Simple Harmonic motion (mass on an ideal spring) Hooke’s Law F = -kx where k is the spring constant, x is displacement from equilibrium. m x=0 equilibrium F x m Physics 153 2006 Simple Harmonic Motion Consider the motion of mass attached to a spring. The force acting on it is -kx. Thus yields -kx = ma x Note this is not a constant acceleration problem because a x depends on x. F = m a x=0 F x m a x Physics 153 2006 SHM cont’d One can see that the motion should repeat itself because of the way a x changes. A good guess of a solution is x = A cos( ! t + " ) A is the amplitude, the maximum displacement from equilibrium. " is the phase constant or phase shift. (All angles are measured in radians. ) Physics 153 2006 Physics 153 2006 Now, lets check that it is a solution. If x = A cos( ! t + " ), then v x = -A ! sin( ! t + " ) a x = -A ! 2 cos( ! t + " ). Plug into the force equation next. Physics 153 2006 -kx = ma x-k A cos( ! t + " ) = ma x = m(-A ! 2 cos( ! t + " ))-k A cos( ! t + " ) = m(-A ! 2 cos( ! t + " )) Canceling like terms yields -k= -m ! 2 Thus x = A cos( ! t + " ) is a solution if and only if ! 2 = k/m Note that ! = 2 " /T as ! T = 2 " determines the period of cosine Physics 153 2006 Physics 153 2006 An object attached to a spring is displaced 0.120m from its equilibrium position and released with zero speed. After .800 s, its displacement is found to be 0.60m on the opposite side and it has passed the equilibrium position and the position of maximum displacement once during this interval. Find the amplitude, period and frequency of this SHM. Physics 153 2006 Defnition oF simple harmonic motion Motion is simple harmonic if and only if it satis¡es the following equation # ! 2 x = a x The solution is x = A cos( ! t + " ). In case of a spring, ! 2 = k/m. ! is the angular frequency measured in units of radians/time; ! = 2 # f. Period is de¡ned to be the time over which the motion repeats. Frequency is the number of cycles per time. Physics 153 2006 Example: Vertical Spring M k The forces acting on the mass are gravity and the spring forces. F y = ma y-ky -mg = ma y Now, as written the equation doesn’t look like simple harmonic motion. Note y measures how much the spring is stretched....
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harmonicwaves - Physics 153 2006 Week 8 Simple Harmonic...

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