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The trajectory of revolution of “Two body” system is circular for comparable mass.

The trajectory of two body system depends on the initial velocities of the bodies and their relative mass. If the mass of the bodies under consideration are comparable, then bodies move around their “center of mass” along two separate circular trajectories. This common point about which two bodies revolve is also known as “barycenter”.

In order to meet the requirement imposed by laws of motion and conservation laws, the motion of two bodies executing circular motion is constrained in certain ways.

Circular trajectory

Since external force is zero, the acceleration of center of mass is zero. This is the first constraint. For easy visualization of this constraint, we consider that center of mass of the system is at rest in a particular reference frame.

Now, since bodies are moving along two circular paths about "center of mass", their motions should be synchronized in a manner so that the length of line, joining their centers, is a constant . This is required; otherwise center of mass will not remain stationary in the chosen reference. Therefore, the linear distance between bodies is a constant and is given by :

Two body system - circular motion

Each body moves around center of mass.

r = r 1 + r 2

Now this condition can be met even if two bodies move in different planes. However, there is no external torque on the system. It means that the angular momentum of the system is conserved. This has an important deduction : the plane of two circular trajectories should be same.

Mathematically, we can conclude this, using the concept of angular momentum. We know that torque is equal to time rate of change of angular momentum,

L t = r × F

But, external torque is zero. Hence,

r × F = 0

It means that “ r ” and “ F ” are always parallel. It is only possible if two planes of circles are same. We, therefore, conclude that motions of two bodies are coplanar. For coplanar circular motion, center of mass is given by definition as :

Two body system - circular motion

Each body moves around center of mass.

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 with respect to time, we have :

m 1 v 1 = m 2 v 2

Now dividing second equation by first,

m 1 v 1 m 1 r 1 = m 2 v 2 m 2 r 2

v 1 r 1 = v 2 r 2

ω 1 = ω 2 = ω s a y

It means that two bodies move in such a manner that their angular velocities are equal.

Two body system - circular motion

Both bodies move with same angular velocity.

Gravitational force

The gravitational force on each of the bodies is constant and is given by :

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

Since gravitational force provides for the requirement of centripetal force in each case, it is also same in two cases. Centripetal force is given by :

F C = m 1 r 1 ω 2 = m 2 r 2 ω 2 = G m 1 m 2 r 2

Angular velocity

Each body moves along a circular path. The gravitational force on either of them provides the centripetal force required for circular motion. Hence, centripetal force is :

m 1 r 1 ω 2 = G m 1 m 2 r 1 + r 2 2

ω 2 = G m 2 r 1 r 1 + r 2 2

Let the combined mass be “M”. Then,

M = m 1 + m 2

Using relation m 1 r 1 = m 2 r 2 , we have :

Questions & Answers

what's the easiest and fastest way to the synthesize AgNP?
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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can anyone tell who founded equations of motion !?
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n=a+b/T² find the linear express
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Moment of inertia of a bar in terms of perpendicular axis theorem
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How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
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Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
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A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
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Torque
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Why are we takingspherical surface area in case of solid sphere
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In all situatuons, what can I generalize?
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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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