lecture 23

lecture 23 - 1 Lecture 23 • Simple harmonic motion •...

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Unformatted text preview: 1 Lecture 23 • Simple harmonic motion • Mass and spring system 2 Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point ) is proportional to the displacement from equilibrium. 3 The motion of a mass on a spring is an example of SHM. x Equilibrium position x y x x m k dt x d dt x d m ma kx F 2 2 2 2 2 ϖ- =- = = =- = ∑ 4 x dt x d 2 2 2 ϖ- = Is a 2 nd order, linear, homogeneous differential equation Its solution is the function x(t) that satisfies equation (1). (1) Guess a solution: ( 29 ( 29 ( 29 ( 29 t x t B t A t x t B t A t x t B t A t x 2 2 2 cos sin sin cos cos sin ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ- =-- =- = + = 5 The guess works! The position of the mass is described by ( 29 t B t A t x ϖ ϖ cos sin + = Where A and B are constants to be determined by the initial conditions. 6 Example: Use the given initial conditions to determine the motion of the mass m....
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lecture 23 - 1 Lecture 23 • Simple harmonic motion •...

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