天体简谐运动

题目

  如图所示，均匀的细圆环半径为$R$，质量为$M$，在中心轴上距环中心$d$处有一质量为$m$的天体，则：
  (1).物体$m$受圆环的万有引力$F$如何？
  (2).若$R>>d$时，则$F$为多大？若细圆环固定，将物体$m$从某一位置释放会如何运动？

程序演示

程序源码

G= 2000;M = 1.0;m = 1.0;R = 10;dt = 0.0001;t = 0;balllist=[];r=[]
scene = canvas(width=300, height=300, 
               background=vector(0.9,0.9,0.9),center=vector(0,2,0))         
ball = sphere(pos=vector(0,1,0), 
              radius = 0.9, color=color.green,make_trail=0.01)
ball.v = vector(0,0,0)
def Fn(r):
    return-(G*M*m*(r)/mag(r)**3)
for N in range(0,360,1):
    balllist.append(sphere(pos=vector(R*cos(N*pi/180),0,R*sin(N*pi/180)),
                        radius = 0.1, color=color.blue))
    r.append(arrow(pos=balllist[N].pos,shaftwidth=0.0001))   
figure=graph(title="v-t图像",width=200, height=200,  
         xtitle="t/(s)",ytitle="速度/（m/s）")
vyt=gcurve(graph=figure,color=color.red)   
while t<1.5:
    t += dt
    rate (1000)
    a=[]   
    for b in range(0,360,1):
        r[b].axis = ball.pos-r[b].pos  
    for q in range(0,360,1):
        a.append(Fn(r[q].axis)/m)
    a_sum = vector(0,0,0)
    for c in range(0,360,1):
        a_sum += a[c]
    ball.v += a_sum*dt
    ball.pos = ball.pos + ball.v*dt
    vyt.plot(t,ball.v.y)