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

Matlab program of location a point with respect a line, matlab program of l...

matlab program of location a point with respect a line

Find out the intensity of the external pressure, Find out the intensity of ...

Find out the intensity of the external pressure: A thick cylinder of 0.5 m external diameter & 0.4 m internal diameter is subjected simultaneously to internal & external press

Two Stage Reciprocating Air Compressors, How does the Two Stage Reciprocati...

How does the Two Stage Reciprocating Air Compressor works?

Computer integrated manufacturing, methods involved in grouping the parts i...

methods involved in grouping the parts into families?

Determine tangential and normal stresses, A rectangular block of material i...

A rectangular block of material is subjected to stresses 65N/mm 2 (tensile) in x- direction and 35 N/mm 2 (tensile) in perpendicular direction. It is accompanied by shear stress

Relation for fin insulated tip and the heat dissipation, Derive the relatio...

Derive the relation for a fin insulated at the tip and the heat dissipation?

Electrical system, Electrical System: Electrical system consists of a batt...

Electrical System: Electrical system consists of a battery, spark plug, lighting system, ignition system, charging system and electricity generator. These components of electrical

Describe about chain-wire and tape gauges, Q. Describe about Chain-wire and...

Q. Describe about Chain-wire and tape gauges? Chain, wire and tape gauges are fixed on structures like bridges and the weight is lowered to touch the water surface. Movable sti

Kinematics in one and two dimensions , A person climbs up a escalator in 60...

A person climbs up a escalator in 60s . When he stands on it , then he is carried in the moving escalator in 30s . How much time will he take in climbing up in a moving escalator?

What is the net angle of twist at the free end, What is the net angle of tw...

What is the net angle of twist at the free end: The stepped steel shaft illustrated in Figure is subjected to a torque (T) at the free end and a torque (2T) in the opposite di

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