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# Testing Phys 8110

Clemson

GPA 3.1

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## Popular in Thermodynamics and Statistical Mechanics

## Popular in Physics 2

This 4 page Class Notes was uploaded by Bipin Notetaker on Tuesday January 19, 2016. The Class Notes belongs to Phys 8110 at Clemson University taught by Prof. Dieter Hartmann in Spring 2016. Since its upload, it has received 54 views. For similar materials see Thermodynamics and Statistical Mechanics in Physics 2 at Clemson University.

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Date Created: 01/19/16

PHYS-8210 CLASSICAL MECHANICS LECTURE-22 BipinSharma (bipins@g.clemson.edu) MihiriHewa Bosthanthirige(mhewabo@g.clemson.edu) Date: 10/14/2015 Minutes: Review of the central force problem E-function of Dissipation Forces Right choice of frame of reference Homework Review of Central Force Problem: We reviewed the central force problem, in which a neutrino moving in its orbit passes through the sun. Consider a shell inside the sun, with the inner and outer radius as r and r+dr respectively, i.e. thickness of the shell is dr. dr Earth The mass of the sun is given by M, radius by R and density by (). Let the orbiting test particle (neutrino) have a given mass m. The potential inside the sun is a function of the distance r of the test particle from the center of gravity inside the sun and is defined by V(r), such that the force acting on the particle is defined as = −∇(). Going back to the shell, the mass of the shell is given by: = 4 . . It is observed that the potential at any point on the shell due to the mass of the sun outside the shell gets cancelled out. Thus the potential at a point which is at a distance r from the center of gravity is () given as = − . Also, it is to be noted that the total mass of the shell with the radius r is given by = ∫ .4 .(). Thus the total potential energy is given by . . ) = 4 ..(− ) E-Function of Dissipative Forces: Inthelastclass,wecametotheconclusionthattheEuler-Lagrangeequationofmotioninthepresence of dissipative forcs ( ), especially Frictioal Force (F ), is given by − = − ̇ ………..(1) The E-function associated with the dissipative forces is given by ∙∙∙ = ̇ − ̇ Therefore, the time derivative of the E-function is given by = ̇ ̇ − = ̈ ̇ + ̇ ̇ − = ̇ − ̇ = ̇ − = − ̇ ̇ ̇ ∴ = − ̇ Choosing the right frame of reference: All the calculations for the equations of motion for a given particle is dependent on the frame of reference that we choose because we consider the particle to be moving relative to that frame of reference. It is very important to choose the right frame of reference. Consider an event in the lab frame. Let a particle be in motion with respect to this frame of reference and let the event take place in the Minkowski spacetime (M ). Let the event be represented by e = X . µ X τ m 0 At the point highlighted in the curve representing the even above, the four velocity is given by = ̅ = ̇ = = = 1 ⃗ ⃗ ⃗ The four momentum is given by = ,and the four acceleration is = = ( ) = ( ) ⃗ ⃗ Where, = Lorentz factor; = (1 − ) 1 1 ∴ ̇ = − (1 − ) −2 = ⃗ 2 In the lab frame,⃗ = ⃗,̇ = ⃗, where, ⃗is the classical (three) acceleration. In this case, 0 ( ) = + ⃗ ⃗ ⃗ The frame of reference where the equations of motion are easiest to calculate in is the rest frame of the particle. In the rest frame, the velocity and acceleration at instantaneous rest position are taken as = and = . Then, = > 0; and = − ⃗ < 0 (space-like acceleration). 0⃗ ⃗ ⃗,⃗,⃗ t m0 lab In the above figure,⃗ is four velocity⃗is three acceleration and⃗is the relativistic momentum. Once we have an idea of ⃗, we can easily derive β and the Lorentz factor γ. Energy of the particle, = , where is the relativistic mass. ⃗ Relativistic three momentum of the particle = ⃗. The four momentum may be written as = = ⃗ = = ⃗ The time derivative of the three momentum is = Ƒ = ⃗ ⃗ Where, s the Lorentz Force. Thus, the four force can be written as = = = ( ̅) + ̅ If the mass does not decay with time, then we can ignore the ̅term since wil be zero. Therefore, = .Now, since the four force is perpendicular to the four velocity, we have: .̅ = 0 ≡ − ⃗ = 0 ⃗ Ƒ ⃗ ⃗ = = = Ƒ⃗ = Ƒ Therefore, the four force is Ƒ = = ⃗ Ƒ⃗ Homework: Make a summary of all the Relativistic Kinematic Equations in a single sheet.

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