一质点同时参与三个简谐振动，它们的振动方程分别为：(x)_(1)=Acos ( (omega t+frac {pi ) (2)} )，(x)_(2)=Acos ( (omega t+frac {7pi ) (6)} )，(x)_(3)=Acos ( (omega t-frac {pi ) (6)} )，其合成运动的运动方程为( )A.x=Acos ( (omega t+frac {3pi ) (2)} )B.x=Acos ( (omega t+frac {5pi ) (6)} )C.x=Acosomega tD.x=0
一质点同时参与三个简谐振动，它们的振动方程分别为：${x}_{1}=Acos\left ( {\omega t+\frac {\pi } {2}} \right )$，${x}_{2}=Acos\left ( {\omega t+\frac {7\pi } {6}} \right )$，${x}_{3}=Acos\left ( {\omega t-\frac {\pi } {6}} \right )$，其合成运动的运动方程为( )
A.$x=Acos\left ( {\omega t+\frac {3\pi } {2}} \right )$
B.$x=Acos\left ( {\omega t+\frac {5\pi } {6}} \right )$
C.$x=Acos\omega t$
D.$x=0$
题目解答
答案
【解析】：
${x}_{1}=Acos(\omega t+\frac {\pi } {2})=-Asin(\omega t)$
${x}_{2}=Acos(\omega t+\frac {7\pi } {6})=-Acos(\omega t+\frac {\pi } {6})$
故${x}_{2}+{x}_{3}=-Acos(\omega t+\frac {\pi } {6})+Acos(\omega t-\frac {\pi } {6})=2Asin(\omega t)sin\frac {\pi } {6}=Asin(\omega t)$
上面这一步是根据和差化积公式：$cos(\alpha -\beta )-cos(\alpha +\beta )=2sin\alpha sin\beta $
合成运动方程为${x}_{1}+({x}_{2}+{x}_{3})=-Asin(\omega t)+Asin(wt)=0$
故本题选：D
解析
${x}_{1}=Acos\left ( {\omega t+\frac {\pi } {2}} \right )=-Asin(\omega t)$
${x}_{2}=Acos\left ( {\omega t+\frac {7\pi } {6}} \right )=-Acos\left ( {\omega t+\frac {\pi } {6}} \right )$
${x}_{3}=Acos\left ( {\omega t-\frac {\pi } {6}} \right )$
步骤 2：利用和差化积公式计算${x}_{2}+{x}_{3}$
${x}_{2}+{x}_{3}=-Acos\left ( {\omega t+\frac {\pi } {6}} \right )+Acos\left ( {\omega t-\frac {\pi } {6}} \right )$
$=2Asin(\omega t)sin\frac {\pi } {6}$
$=Asin(\omega t)$
步骤 3：计算合成运动方程
合成运动方程为${x}_{1}+({x}_{2}+{x}_{3})$
$=-Asin(\omega t)+Asin(\omega t)$
$=0$