Download - Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field
![Page 1: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/1.jpg)
Central Force
Umiatin,M.Si
![Page 2: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/2.jpg)
• The aim : to evaluate characteristic of motion under central force field
![Page 3: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/3.jpg)
A. Introduction
• Central Force always directed along the line connecting the center of the two bodies
• Occurs in : motion of celestial bodies and nuclear interaction
![Page 4: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/4.jpg)
Central Force Motion as One Body Problem
• Suppose isolated system consist two bodies and separated a distance r = |r| with interaction between them described by a central force F(r), we need six quantities used to describe motion of those particle :
![Page 5: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/5.jpg)
![Page 6: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/6.jpg)
Method 1 :
• To describe those, we need six quantities ( three component of r1 and three component of r2). The equation of motion of those particle are :
• If F(r) > 0: repusive , F(r) <0 : Attractive. Coupled by :
![Page 7: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/7.jpg)
Method 2
• Describe a system using center mass (R) and relative position (r).
• R describes the motion of the center of mass and r describes the relative motion of one particle with respect to the other
![Page 8: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/8.jpg)
No external forces are acting on the system, so the motion of the center of mass is uniform translational motion. R** = 0.
![Page 9: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/9.jpg)
• Where reduced mass define by :
• Two bodies problem has been simplified into one body problem.
![Page 10: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/10.jpg)
![Page 11: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/11.jpg)
• Solve the equation of motion :
• The center of mass moves with uniform velocity :
• By choosing the initial condition, vo, to, Ro = 0, the origin of coordinate coincides with center of mass R.
![Page 12: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/12.jpg)
• So the position of m1 and m2 which measured from center of mass :
![Page 13: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/13.jpg)
![Page 14: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/14.jpg)
If m2 >> m1, then reduced mass:
The eq of motion :
Become :
![Page 15: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/15.jpg)
• Hence the problem can be treated as a one body problem. Thus, whenever we use mass m instead of µ, we are indicating that the other mass is very large, whereas the use of µ indicates that either the two masses are comparable.
![Page 16: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/16.jpg)
B. General properties of Central Force
1. Central Force is Confined to a PlaneIf p is the linear momentum of a particle of mass µ, the torque τ about an axis passing through the center of force is :
![Page 17: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/17.jpg)
• If the angular momentum L of mass µ is constant, its magnitude and direction are fixed in space. Hence, by definition of the cross product, if the direction of L is fixed in space, vectors r and p must lie in a plane perpendicular to L. That is, the motion of particle of mass µ is confined to a plane that is perpendicular to L.
![Page 18: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/18.jpg)
![Page 19: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/19.jpg)
• As we the force acting at body is central force, three dimensional problem can be reduced into two dimensional. Using polar coordinate system :
![Page 20: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/20.jpg)
2. Angular Momentum and Energy are Constant
The angular momentum of a particle of mass µ at a distance r from the force center is :
![Page 21: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/21.jpg)
![Page 22: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/22.jpg)
• Since there are no dissipative systems and central forces are conservative, the total energy is constant :
![Page 23: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/23.jpg)
3. Law of Equal Areas
Consider a mass µ at a distance r(θ) at time t from the force center O :
![Page 24: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/24.jpg)
• Subtituting
![Page 25: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/25.jpg)
![Page 26: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/26.jpg)
C. Equation of Motion
From the previous description :
• If we know V(r), these equations can be solved for θ(t) and r(t). The set [θ(t), r(t)] describes the orbit of the particle.
![Page 27: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/27.jpg)
• Solve the equation, we find :
• We will get t(r) then inverse r(t). But we are interested in the equation of the path in term r and θ
![Page 28: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/28.jpg)
• We may write :
• And subtitute :
• Then :
![Page 29: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/29.jpg)
Suppose the Force is F(r) = Krn
• K = constant• If n = 1 the solution is motion of
harmonic oscillator• If n = -2 , eq : coulomb and gravitation
force
![Page 30: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/30.jpg)
Other method : Use Lagrangian
• Lagrangian of the system :
• We find
![Page 31: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/31.jpg)
• To simplify, use other variable, for example : u in which = 1/r
![Page 32: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/32.jpg)
• Next find
• Therefore :
![Page 33: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/33.jpg)
• We can transform into :
![Page 34: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/34.jpg)
Example
1. Find the force law for a central force field that allow a particle to move in logarithmic spiral orbit given by (k and α are constant) :
![Page 35: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/35.jpg)
• Solution :First determine :
![Page 36: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/36.jpg)
• Now determine :
![Page 37: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/37.jpg)
2. Find r(t) and θ(t) !Solution :
![Page 38: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/38.jpg)
![Page 39: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/39.jpg)
3. What is the total energy ?
• Solution :
![Page 40: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/40.jpg)
• We know that
![Page 41: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/41.jpg)
D. Planetary Motion
The equation for the path of a particle moving under the influence of a central force whose magnitude is inversely proportional to the distance between the particle can be obtain from :
![Page 42: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/42.jpg)
• If we define the origin of θ so that the minimum value of r occurs at θ = 0, so
![Page 43: Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field](https://reader036.vdocument.in/reader036/viewer/2022062600/5a4d1b957f8b9ab0599c3404/html5/thumbnails/43.jpg)