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

Basic concept of servicing and maintenance, Basic Concep: Proper care and ...

Basic Concep: Proper care and maintenance are essential for trouble free operation and optimum performance of the motorcycle. A quality service at regular intervals ensures that e

Agriculture, i am thinking that neglecting the foot gears and put hand gear...

i am thinking that neglecting the foot gears and put hand gears instead of it in two wheelers .so what is the procedure

Limit of proportionality, Limit of proportionality: Limit of proporti...

Limit of proportionality: Limit of proportionality is the stress at which the stress - strain diagram ceases to be a straight line that is, stress at which the extension ceas

Kinematics, how do you find the projectile motion or equation?

how do you find the projectile motion or equation?

Evaluate the natural frequency, (a) Evaluate the natural frequency of syste...

(a) Evaluate the natural frequency of system shown in figure and show that it will be equal to (b) calculate equivalent stiffness of two springs connected in parallel.

Fourth gear position - shifting of gear, Fourth Gear Position: When g...

Fourth Gear Position: When gear M4 on the main shaft is in mesh with gear C4 on the countershaft then this is known as fourth gear position. The free gear M4 rotates with the

Thermodynamics, define the first and second law of thermodynamics.

define the first and second law of thermodynamics.

I.c.engine, compression ignition engine fore stork with neat daysgram

compression ignition engine fore stork with neat daysgram

Lathe - milling and grinding machine, Define O.R. and what are the applicat...

Define O.R. and what are the applications in Industry ? A firm uses lathe, milling and Grinding machine to make two machine parts. The table below show the machining time requir

Evaluate the normal and tangential stresses, Evaluate the normal and tangen...

Evaluate the normal and tangential stresses: Q. 20: At the point in stressed body, the principal stresses are σ x = 80 kN/m 2 and σ y = 40  kN/m 2 . Determine the norma

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