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

Wave motion, Wave motion takes place all throughout nature, from the waves...

Wave motion takes place all throughout nature, from the waves of the ocean to the electromagnetic waves that comprise visible light. Though different types of waves involve differe

Welding transformer, Welding transformer The resistance welding transfo...

Welding transformer The resistance welding transformers step down the line tune of 2 to 20 V. The primary is connected to the line supply and is made from edge bent copper stri

Boiler, boiler performance.

boiler performance.

Cad, nc & cnc machines

nc & cnc machines

Compression ratio and mean effective pressure, Compressio n Ratio: It...

Compressio n Ratio: It is calculated as shown below Mea n Effective Pressure (P m   o r P me f ): Mean effective pressure is the hypothetical consta

Varignon theorem - mechanics, Varignon theorem: State Varignon's theo...

Varignon theorem: State Varignon's theorem. How it helps in determining of the moments? In which type of condition is it used? Sol.: Varignon's theorem also called Law

Motorcycle pulls to one side motorcycle problem, If Motorcycle Pulls to One...

If Motorcycle Pulls to One Side Causes of Problem Remedy Misaligned shock Absorber Front and rear wheels are not alig

Evaluate the actual over all efficiency of the turbine, (a) An engine uses ...

(a) An engine uses 6.5 Kg of oil per hr of a calorific value 30,000 kJ / kg. If the B. P. of the engine is 22KW and mechanical efficiency 85%. Calculate (i) Brake thermal eff

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