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Since no external force is applied, the subsequent motion due to internal gravitational force does not change the position of center of mass in accordance with second law of motion. The bodies simply move towards each other such that center of mass remains at rest.

The two bodies move along a straight line joining their centers. The line of motion also also passes through center of mass. This has one important implication. The plane containing one body and “center of mass” and the plane containing other body and “center of mass” are same. It means that motions of two bodies are “coplanar” with line joining the centers of bodies and center of mass. The non-planar motions as shown in the figure below are not possible as motion is not along the line joining the centers of two bodies.

Two body system

The motion of two body system can not be non-planar.

Let suscripts "1" and "2" denote two bodies. Also, let “ r 1 ”, “ v 1 ”, “ a 1 ” and “ r 2 ”, “ v 2 ”, “ a 2 ” be the magnitudes of linear distance from the center of mass, speeds and magnitudes of accelerations respectively of two bodies under consideration. Also let “center of mass” of the system is the origin of reference frame. Then, by definition of center of mass :

Two body system

Center of mass is the origin of reference frame.

r c m = - m 1 r 1 + m 2 r 2 m 1 + m 2

But “center of mass” lies at the origin of the reference frame,

r c m = - m 1 r 1 + m 2 r 2 m 1 + m 2 = 0

m 1 r 1 = m 2 r 2

Taking first differentiation of position with respect to time, we have :

v c m = - m 1 v 1 + m 2 v 2 m 1 + m 2 = 0

m 1 v 1 = m 2 v 2

Taking first differentiation of velocity with respect to time, we have :

a c m = - m 1 a 1 + m 2 a 2 m 1 + m 2 = 0

m 1 a 1 = m 2 a 2

Considering only magnitude and combining with Newton’s law of gravitation,

F 12 = F 21 = G m 1 m 2 r 1 + r 2 2

Since distance of bodies from center of mass changes with time, the gravitational force on two bodies is equal in magnitude at a given instant, but varies with time.

Newton’s second law of motion

We can treat “two body” system equivalent to “one body” system by stating law of motion in appropriate terms. For example, it would be interesting to know how force can be related to the relative acceleration with which two bodies are approaching towards each other. Again, we would avoid vector notation and only consider the magnitudes of accelerations involved. The relative acceleration is sum of the magnitudes of individual accelerations of the bodies approaching towards each other :

a r = a 1 + a 2

According to Newton's third law, gravitational force on two bodies are pair of action and reaction and hence are equal in magnitude. The magnitude of force on each of the bodies is related to acceleration as :

F = m 1 a 1 = m 2 a 2

m 1 a 1 = m 2 a 2

We can note here that this relation, as a matter of fact, is same as obtained using concept of center of mass. Now, we can write magnitude of relative acceleration of two bodies, “ a r ”, in terms of individual accelerations is :

a r = a 1 + m 1 a 1 m 2

a r = a 1 m 1 + m 2 m 2

a 1 = m 2 a r m 1 + m 2

Substituting in the Newton’s law of motion,

F = m 1 a 1 = m 1 m 2 a r m 1 + m 2

F = μ a r


μ = m 1 m 2 m 1 + m 2

Reduced mass

The quantity given by the expression :

Questions & Answers

List the application of projectile
Luther Reply
How can we take advantage of our knowledge about motion?
Kenneth Reply
pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
what are scalars
Abdhool Reply
show that 1w= 10^7ergs^-1
Lawrence Reply
what's lamin's theorems and it's mathematics representative
Yusuf Reply
if the wavelength is double,what is the frequency of the wave
Ekanem Reply
What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
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water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
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Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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