[Mechanics] Simple Harmonic Motion

2021-10-21 10:00 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

Consider the position function $x(t)%20%3D%20A%20%5Ccos%7B(%5Comega%20t%20%2B%20%5Cphi)%7D$ . Show that this function solves the differential equation

$%5Cfrac%7Bd%5E2%20x%7D%7Bdt%5E2%7D%2B%5Cfrac%7Bkx%7D%7Bm%7D%20%3D%200$

and determine the value of $%5Comega$ in terms of $k%2Cm$ .

【Solution】

Take two time derivatives:

$%5Cfrac%7Bdx%7D%7Bdt%7D%3D%20-%20%5Comega%20A%20%5Csin%7B(%5Comega%20t%20%2B%20%5Cphi)%7D%20$

$%5Cfrac%7Bd%5E2%20x%7D%7Bdt%5E2%7D%20%3D%20-%20%7B%5Comega%7D%5E%7B2%7D%20A%20%5Ccos%7B(%5Comega%20t%20%2B%20%5Cphi)%7D$

Consequently,

$-%20%7B%5Comega%7D%5E%7B2%7D%20A%20%5Ccos%7B(%5Comega%20t%20%2B%20%5Cphi)%7D%20%2B%20%5Cfrac%7Bk%7D%7Bm%7D%20A%20%5Ccos%7B(%5Comega%20t%20%2B%20%5Cphi)%7D%20%3D%200%20$

$%5Cleft(%5Cfrac%7Bk%7D%7Bm%7D-%7B%5Comega%7D%5E%7B2%7D%5Cright)A%20%5Ccos%7B(%5Comega%20t%20%2B%20%5Cphi)%7D%20%3D%200$

Divide away $A%20%5Ccos%7B(%5Comega%20t%20%2B%20%5Cphi)%7D%20$ :

$%5Cleft(%5Cfrac%7Bk%7D%7Bm%7D-%7B%5Comega%7D%5E%7B2%7D%5Cright)%20%3D%200$

Therefore, $%5Comega%20%3D%20%5Cpm%20%5Csqrt%7B%5Cfrac%7Bk%7D%7Bm%7D%7D%20$ .

This quantity is the angular frequency of the wave. Mathematically speaking, it is the eigenvalue of the differential equation.