Symmetric central force, Mechanical Engineering

Symmetric Central Force

A central force is spherically if the magnitude of the force does not depend on the direction (angles θ, or Ø) of the particle but only on distance r from the center of the force, that is F is spherically symmetric if,


F = F ( r ) er

A spherically symmetric force is conservative, conversely, if a central force during a small displacement d l of the particle:

dW = F . d l = ( F ( r ) er ) . ( er dr + eθ r dθ)

= F ( r ) dr

where we used dl in polar co-ordinates. The work done in moving the particle from r1 to r2, therefore, is

809_download.png

where, G ( r ) is the integral function of F ( r ); that is, F ( r ) = dG ( r )/dr

The work done thus depends only on the end co-ordinates and not on the path followed. Force F ( r ) er is therefore conservative.

G ( r ), in fact expresses the negative of potential energy function U ( r ):

W = - ( U ( r2 ) - U ( r1 ))

For example, the gravitational and electrostatic forces are spherically symmetric central forces. These are expressed as

1718_download (1).png 

We discussed these forces earlier; the potential energy of interaction for the two particles in case of above forces is:

1288_download (2).png 

The central forces (like gravitational or electrostatic) which vary as (1/r2) are called inverse square forces.

The conservation of energy principal for a particle moving in a spherically symmetric central force is expressed as,

½ mv2 + U ( r ) = E , constant

430_download (3).png 

On the other hand, the angular momentum of the particle is also a constant of motion:

2138_download (4).png 

Hence we get

2421_download (5).png

1507_download (7).png 

represents an effective potential energy. (Remember, L2/2 mr2 is really a part of kinetic energy coming from transverse motion of the particle.)

If the particle motion is a one-dimensional motion along the radial direction under effective potential energy function U' ( r ). The entire effect of transverse motion of the particle is incorporated in the potential energy as additional L2/2 mr2 term.

The term L2/2mr2 is sometimes referred to as the 'centrifugal' potential energy. This is because the corresponding force is,

1686_download (6).png 

which is same as centrifugal force mr ω2 in a co-ordinate frame rotating with instantaneous angular velocity ω = dθ/dt.

Posted Date: 9/18/2012 1:31:24 AM | Location : United States







Related Discussions:- Symmetric central force, Assignment Help, Ask Question on Symmetric central force, Get Answer, Expert's Help, Symmetric central force Discussions

Write discussion on Symmetric central force
Your posts are moderated
Related Questions
Q. Design Loads and Load Combinations? The Designer shall determine the following loads and specify them on the Data Sheet. Design loads are defined and classified as follows:

Figure shows a reinforced concrete framed building subjected to earthquake ground motion. The floor is rigid with the mass of each floor is shown in the figure. Formulate the equat


Explain Degree of Freedom - Jig and Fixture The complete location of components inside a jig or fixtures can be understood with the help of figure. The work piece is a

compressor’s volumetric efficiency can also depend on the compressor’s ratio.


Base Circle: this is the smallest circle tangent to the cam profile. Trace Point: this is theoretical point on the follower whose motion explained the movement of the foll

Batch Furnaces: An insulated, the heating system, chamber for placing the job, and a door or various doors for placing the job in place are the necessity of these furnaces. T

In this assignment you will apply what you have learned in previous training. Consider the part in figure, which constructed from a solid tube. The bracket is supported at surface

A steel member supported by a tie bar has a load of 15 000 lbs on it. If the force on the steel member of 1.2 in 2 cross section is 14,180 lbs and the force on the tie bar of diam