harmonic

# harmonic - Physics 153 2006 by Don Witt Periodic Motion...

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Physics 153 2006 by Don Witt Periodic Motion Periodic Motion is motion that repeats itself. Period is defined to be the time over which the motion repeats. Frequency is the number of cycles per time. Amplitude is the maximum displacement from equilibrium.

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Physics 153 2006 by Don Witt Hooke’s Law F = -kx where k is the spring constant, x is how much from equilibrium.
Physics 153 2006 by Don Witt Simple Harmonic Motion Consider the motion of mass with a spring force actting on it -kx = ma x where F = ma was used. First note this is not a constant acceleration problem Because a x depends on x.

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Physics 153 2006 by Don Witt SHM cont’d One also see the motion should repeat itself because of the way a x changes. A good guess of a solution is x = A cos( ω t + φ ) A is the amplitude that is maximum displacement from equilibrium. φ is the phase constant or phase shift. (All angles are measured in radians. )
Physics 153 2006 by Don Witt Now, lets check that the above expression is a solution. If x = A cos( ω t + φ ), then v x = -A ω sin( ω t + φ ), and a x = -A ω 2 cos( ω t + φ ). Plug into the force equation, next.

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Physics 153 2006 by Don Witt -kx = -k A cos( ω t + φ ) = ma x -k A cos( ω t + φ ) = ma x = m(-A ω 2 cos( ω t + φ )) -k A cos( ω t + φ ) = m(-A ω 2 cos( ω t + φ )) Canceling like terms yields x = A cos( ω t + φ ) is solution if and only if ω 2 = k/m
Physics 153 2006 by Don Witt Definition of simple harmonic motion Motion is simple harmonic if and only if it satisfies the following equation −ω 2 x = a x The solution is x = A cos( ω t + φ ). In case of a spring ω 2 = k/m. ω is the angular frequency measured in units of radians/time. Period is defined to be the time over which the motion repeats. Frequency is the number of cycles per time.

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Physics 153 2006 by Don Witt Example Vertical Spring M k The forces acting on the mass are gravity and the spring forces. F = ma -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.
Physics 153 2006 by Don Witt M k To make it easy to solve this problem, find where the net force zero in other words where the system is in equilibrium F = -ky -mg = 0. This is where y= -mg/k . Now,measure everything from this equilibrium position. So pick a new coordinate, namely, y new = y + mg/k

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Physics 153 2006 by Don Witt Plug y new = y + mg/k into -ky -mg = ma y yields -ky -mg = -k( y new - mg/k) -mg = ma y -k y new = ma y . Finally, since mg/k is constant a y =a ynew . -k y new = ma y new
2006 by Don Witt Finally, this means a vertical spring is simple harmonic motion but the displacement must be measure from the equilibrium position. The angular frequency is given by

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harmonic - Physics 153 2006 by Don Witt Periodic Motion...

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