Three particle system, Mechanical Engineering

Assignment Help:

Three Particle System

Suppose we have two particles of masses m1 and m2 already fixed in space at distance r12 from each other. Let us bring in a third particle of mass m3, from ∞ to some point P near the first two particles, so that m3 finally is at distance r13 from m1 and at distance r23 from m2.


Now, at any instant, there are two forces acting on m3, viz. the gravitational force F31 due to m1 and F32 due to m2. The total work done in moving m3 to point P is given by,

742_download.png 

Note that the two forces act independently of each other along respective radial directions. That is, for example, we have
938_download (1).png 

Note that the two forces act independently of each other along respective radial directions. That is, for example, we have
595_download (2).png 

where and dr in the above integral refer to distances along the radial direction joining particles 1 and 3, at time t. Similarly, we get

2129_download (3).png 

For conservative forces, the work done is interpreted as the negative change in potential energy. Hence, the increase in gravitational potential energy of the system by joining of third particle is (-W3). The total potential energy of three-particle system becomes,

U = U12 + ( -W3 )

156_download (5).png 

Thus, the total potential energy of the system is the sum of potential energies of each pair of particles taken independently.

Remember that ( -W3 ) is not the potential energy 'of mass m3'; it is the sum of potential energies of masses (m1 and m3)and masses (m2 and m3).

If m3 = 1 (unit mass), we define the gravitational field at point P due to masses m1 and m2 as the net force acting on unit mass at P.

2150_download (6).png 

where we are now writing r1 and r2 as the position vectors of point P relative to masses m1 and m2. [That is, in fact, r1 ≡ r31and r2 = r32].

Gravitational potential at point P due to masses m1 and m2 gives the change in potential energy of the system when a unit mass is added to the system at point P. That is, potential ØP at P is the value of ( -W3 ) from m3 = 1 (unit mass).

1146_download (7).png 

where r1 and r2 denote distances of P from m1 and m2.


Related Discussions:- Three particle system

Occupational health problems, Occupational Health Problems Various he...

Occupational Health Problems Various health problems have been found to be rooted in type of work, equipment and Workplace. It was eighteenth century while definite observati

Concept of continum, in concept of continum what is the meaning of upper cu...

in concept of continum what is the meaning of upper cure

Calculate the pressure using density , An open test tube at 293 K is filled...

An open test tube at 293 K is filled at the bottom with 24.2 cm of Hg, and 11.2 cm of water is placed above the Hg. Calculate the pressure at the bottom of the test tube if the atm

Illustrate the design process and application of cad, Describe the use of c...

Describe the use of computer in : (i) Design (ii) Manufacturing Illustrate the design process and application of CAD.

Explain the information phase screw conveyor with auger, Information phase ...

Information phase of screw conveyor with auger: Steps: To secure all and complete information and gather facts and from best sources.Our project is selected based on the cu

Thermodynamics, what is the difference between point function and path func...

what is the difference between point function and path function?

Compound BArs, three vertical, parallel straight rods, each 2.6m in length ...

three vertical, parallel straight rods, each 2.6m in length have their upper ends firmly secured and their lower supporting a rigid horizontal bar with a weight of 8KN, which cause

What is extrusion defects, Q. What is Extrusion defects? • Surface crac...

Q. What is Extrusion defects? • Surface cracking - When the surface of an extrusion splits. This is often caused by the extrusion temperature, friction, or speed being too high

Equilibrium of rigid bodies, Ask question #MinimumThe man is pulling a load...

Ask question #MinimumThe man is pulling a load of 4kg with one arm held as shown. Determine the force FH this exerts on the humerus bone H, and the tension developed in the biceps

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd