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

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Physics 153 2006 by Don Witt Hooke’s Law F = -kx where k is the spring constant, x is how much from equilibrium.
Background image of page 2
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.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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. )
Background image of page 4
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.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 10
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
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/20/2009 for the course PHYS physics 10 taught by Professor Goatman during the Spring '08 term at UBC.

Page1 / 77

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

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online