class12 - 1.7 2D oscillations Consider the motion of a...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: 1.7 2D oscillations Consider the motion of a particle with two degrees of freedom. Take the restoring force to be proportional to the distance of the particle from a force center located at the origin, and to be directed toward the origin: F =- k r (113) which can be written in polar coordinates: F x =- kr cos =- kxF y =- kr sin =- ky (114) with equations of motion x + 2 x = 0 (115) y + 2 y = 0 (116) As before, = k/m . The solutions are x ( t ) = A cos( t- ) (117) y ( t ) = B cos( t- ) (118) The motion is one of SHO in each of the two directions: both having the same frequency but possibly two different amplitudes and phases. We can obtain the equation for the path y ( t ) = B cos[ t- + ( - )] (119) = B cos( t- ) cos( - )- B sin( t- ) sin( - ) (120) Define - at note that cos( t- ) = x/A , y = B A x cos - B 1- x 2 A 2 sin (121) or Ay- Bx cos =- B A 2- x 2 sin (122) squaring gives A 2 y 2- 2 ABxy cos + B 2 x 2 cos 2 = A 2 B 2 sin 2 - B 2 x 2 sin 2 (123) and B 2 x 2- 2 ABxy cos + A 2 y 2 = A 2 B 2 sin 2 (124) If we set = / 2 this gives the equation of an ellipse x 2 A 2 + y 2 B 2 = 1 , = / 2 (125) where in the special case of A = B we have circular motion. Another special case occurs for = 0, B 2 x 2- 2 ABxy + A 2 y 2 = 0 , = 0 (126) 14 Factoring ( Bx- Ay ) 2 = 0 (127) a straight line, y = xB/A . In the more general case of two-dimensional oscillations, the angular frequencies for the motion in the x and y directions are not equal, so that x ( t ) = A cos( x t- ) (128) y ( t ) = B cos( y t- ) (129) The paths of motion are called...
View Full Document

Page1 / 4

class12 - 1.7 2D oscillations Consider the motion of a...

This preview shows document pages 1 - 3. Sign up to view the full document.

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