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
Find distance through which the body will penetrates: A body having mass 25kg falls on the ground from height of 19.6m. The body penetrates into ground. Find distance through

Determine the slopes at free end - cantilever beam: A cantilever beam of length l carries a point load W at a distance l  /4 from free end. Determine the slopes at free end

equivalent evaporation of boiler..?

What is the total value added by the new workshop in Year 1 of the 5-year plan? Assuming that the average quarterly demand is maintained over the remaining years of the 5-year p

show that energy is property of system

Is it possible for the steady state temperature profile in a plane wall to be nonlinear at steady state? If yes, provide one example. Solution: Steady state temperature profile

(a) Derive an expression for the otto cycle to determine the Mean Effective Pressure and efficiency. (b)The two engines are to perform on diesel and otto cycle with the followin

Illustrate the steady state error for different types of systems. Check the stability of the system using Routh criterion S 6 + 4S 5 + 3S 4 - 16S 2 - 64S - 48 = 0

Space mass and weight: The geometric region occupied by bodies called as space. When body changes its position w. r.t. the other bodies, then body is known to be in motion.

Scope of  the angle  radial  drilling  machine  There  are  many  scope  of  the  angle  radial  drilling  machine  like  as:  we  shell produce a hole at any angle with help