Central Forces
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Transcript of Central Forces
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CENTRAL FORCES
AIM
To simulate the orbit of a particle moving under a central potentialof the form V=-(k/r)
THEORY
Our ultimate goal is to solve the equations of motion for two
masses m1 and m2 subject to a central force acting between them.
When you considered this problem in introductory physics, you
assumed that one of the masses was so large that it effectivelyremained at rest while all of the motion belonged to the other object.
This assumption works fairly well for the Earth orbit in garound the Sun
or for a satellite orbiting around the Earth, but in general we are going
to have to solve for the motion of both objects. In the introduction, we
defined a central force to satisfy two characteristics. We can now write
turn these descriptions of the characteristics into equations:
(a) a central force depends only on the separation between the two
bodies
f21 = f1
Since a central force is a conservative force, we can define a potential
energy. If there are no other forces, then energy is conserved and we
can write this in an equation:
KE + PE = E = constant
m r2
+ m r2
2+ V(r) = E .
If we use the angular momentum expression, = L/mr2, this becomes:
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m r2
+ m r2
(L/mr2)
2+ V(r) = E = m r
2+ L
2/mr
2+ V(r) .
If we treat the term L2
/mr2
as an additional potential energy (since it
only depends on r), we can define an effective potential energy: V(r)
= V(r) + L2 /2mr2 . We now have an equation that is essentially one-
dimensional:
m r2
+ V(r) = E = constant .
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FLOW CHART
OUTPUT
START
READ INPUTS
IF GRAPHICS ERROR STOP.
OR
DO CALCULATIONS
END
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PROGRAM
#include
#include
#include
#include
#include
#include
void main()
{
float theta,dtheta,h,f1,f2,f3,dr,r,rmax=1.8,rmin=0.7,y,x;
float fn(float r);
int xmid,ymid,n,i,j,errorcode;
clrscr();
errorcode=graphresult();
if(errorcode!=grOk)
{
cout
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int gdriver=DETECT,gmode;
initgraph(&gdriver,&gmode,"");
xmid=getmaxx()/2;
ymid=getmaxy()/2;
line(xmid-(0.75*xmid),ymid,xmid+(0.75*xmid),ymid);
line(xmid,ymid-(0.75*ymid),xmid,ymid+0.75*ymid);
dr=(rmax-rmin)/200;
h=dr/100;
r=rmin;
theta=0;
dtheta=0;
for(n=1;n
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x=r*cos(theta);
putpixel(xmid+0.7*xmid*x/rmax,ymid-0.8*ymid*y/rmax,10);
putpixel(xmid+0.7*xmid*x/rmax,ymid+0.8*ymid*y/rmax,10);
}
getch();
closegraph();
}
float fn(float r)
{
float j=1,m=1,k=1,E=-0.4,a,b;
a=2*m*E+2*m*k/r-(j*j)/(r*r);
b=sqrt(a);
return(j/(r*r*b));
}
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CHART
