When two bodies move along a line, there is a special system of coordinates in w
ID: 1330663 • Letter: W
Question
When two bodies move along a line, there is a special system of coordinates in which the momentum of one body is equal and opposite to that of the other. That is, the total momentum of the two bodies is zero. This frame of reference is called the center-of-mass system (abbreviated CM). Let the bodies have masses m1 and m2 and be moving at speeds v1 and V2. Use conservation of momentum and Galileian relativity to show that the CM system is moving at a velocity given by (There is a faster way using the definition of center of mass. Explain.)Explanation / Answer
The definition of center of mass is given " The centre of mass of a system of particle is that point lying within the boundary of system where the entire mass appears to be concentrated"
Now let us determine the co-ordinates of the center of mass of two bodies .Consider a co-ordinate system with x-axis coinciding with the line joining the centers of the two bodies.
Let O be the origin of the co-ordinste system
Let x1 be the co-ordinate of mass m1
x2 be the co-ordinste of mass m2
Xcm be the co-ordinate of center of mass at P
r1 be thedistance from m1 to P and r2 is the distnce from P to m2 and r is the distance between m1and m2
We can give that
r =x2 -x1
r1 =xcm -x1
r2 =x2 -xcm
Now substituting in the equation m1r1 =mr2 and simplifiying we get
Xcm =m1x1+m2x2/(m1+m2)-----(1)
Now let total mass of the two bodies m=m1+m2
Mxcm =m1x1+m2x2------(2)
Let the two bodies moving in the +x -direction
The small chang in position is Deltax1 and Deltax2 and position of center of mass Deltaxcm and time interval is Deltat
M(Xcm +Deltaxcm) =m1(x1+deltax1)+m2(x2+deltax2)----(3)
From the equation (2) and (3) we get
M.Xcm =m1deltax1+m2Deltax2
dividing by deltat we get
MVcm =m1v1+m2v2
Vcm =m1v1+m2v2/(m1+m2)
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