Classical mechanics 3e by herbert goldstein solution manual, Exercises for Physics LATEST2023 GRADED A+ Classical mechanics 3e by herbert goldstein solution manual Physics Riphah International Univeristy (RIU) 149 pag. Document shared on https://www

Classical mechanics 3e by herbert goldstein solution manual, Exercises for Physics LATEST2023 GRADED A+ Classical mechanics 3e by herbert goldstein solution manual Physics Riphah International Univeristy (RIU) 149 pag. Document shared on Downloaded by: winnie-mumbi-1 () Classical mechanics 3e by herbert goldstein solution manual, Exercises for Physics LATEST2023 GRADED A+ Goldstein Classical Mechanics Notes Michael Good May 30, 2004 1 Chapter 1: Elementary Principles Mechanics of a Single Particle Classical mechanics incorporates special relativity. ‘Classical’ refers to the con- tradistinction to ‘quantum’ mechanics. Velocity: Linear momentum: dr v = . dt Force: p = mv. dp F = . dt In most cases, mass is constant and force is simplified: Acceleration: d dv F = (mv) = m dt dt d 2 r = ma. a = dt2 . Newton’s second law of motion holds in a reference frame that is inertial or Galilean. Angular Momentum: L = r ×p. Torque: T = r ×F. Torque is the time derivative of angular momentum: 1 Document shared on Downloaded by: winnie-mumbi-1 () Classical mechanics 3e by herbert goldstein solution manual, Exercises for Physics LATEST2023 GRADED A+ Work: dL T = . dt W12 = ∫ 2 F · dr. 1 In most cases, mass is constant and work simplifies to: W12 = m ∫ 2 dv ∫ 2 · vdt = m 1 dt 1 dv ∫ 2 v · dt = m dt 1 v · dv W = m (v 2 − v 2 ) = T − T Kinetic Energy: 12 2 2 T = 1 2 1 mv 2 2 The work is the change in kinetic energy. A force is considered conservative if the work is the same for any physically possible path. Independence of W12 on the particular path implies that the work done around a closed ciruit is zero: I F · dr = 0 If friction is present, a system is non-conservative. Potential Energy: F = −∇V (r). The capacity to do work that a body or system has by viture of is position is called its potential energy. V above is the potential energy. To express work in a way that is independent of the path taken, a change in a quantity that depends on only the end points is needed. This quantity is potential energy. Work is now V1 −V2. The change is -V. Energy Conservation Theorem for a Particle: If forces acting on a particle are conservative, then the total energy of the particle, T + V, is conserved. The Conservation Theorem for the Linear Momentum of a Particle states that linear momentum, p, is conserved if the total force F, is zero. The Conservation Theorem for the Angular Momentum of a Particle states that angular momentum, L, is conserved if the total torque T, is zero. 2 Document shared on Downloaded by: winnie-mumbi-1 () Classical mechanics 3e by herbert goldstein solution manual, Exercises for Physics LATEST2023 GRADED A+ Mechanics of Many Particles Newton’s third law of motion, equal and opposite forces, does not hold for all forces. It is called the weak law of action and reaction. Center of mass: Σ miri Σ miri R = Σ = . mi M Center of mass moves as if the total external force were acting on the entire mass of the system concentrated at the center of mass. Internal forces that obey Newton’s third law, have no effect on the motion of the center of mass. F (e) d 2R ≡ M dt2 = Σ F (e) . i Motion of center of mass is unaffected. This is how rockets work in space. Total linear momentum: Σ dri dR P = mi dt = M dt . i Conservation Theorem for the Linear Momentum of a System of Particles: If the total external force is zero, the total linear momentum is conserved. The strong law of action and reaction is the condition that the internal forces between two particles, in addition to being equal and opposite, also lie along the line joining the particles. Then the time derivative of angular momentum is the total external torque: dL = N(e) . dt Torque is also called the moment of the external force about the given point. Conservation Theorem for Total Angular Momentum: L is constant in time if the applied torque is zero. Linear Momentum Conservation requires weak law of action and reaction. Angular Momentum Conservation requires strong law of action and reaction. Total Angular Momentum: Σ Σ L = ri ×pi = R ×Mv + r ′ ×p′ . i i i i 3 i Document shared on Downloaded by: winnie-mumbi-1 () Classical mechanics 3e by herbert goldstein solution manual, Exercises for Physics LATEST2023 GRADED A+ Total angular momentum about a point O is the angular momentum of mo- tion concentrated at the center of mass, plus the angular momentum of motion about the center of mass. If the center of mass is at rest wrt the origin then the angular momentum is independent of the point of reference. Total Work: W12 = T2 − T1 where T is the total kinetic energy of the system: T = 1 Σ m v 2 . Total kinetic energy: 2 i i i T = 1 Σ m v 2 = 1 Mv2 + 1 Σ m v ′ 2 . 2 i i 2 i 2 i i i Kinetic energy, like angular momentum, has two parts: the K.E. obtained if all the mass were concentrated at the center of mass, plus the K.E. of motion about the center of mass. Total potential energy: V = Σ 1Σ Vi + 2 i i,j i Vij . j If the external and internal forces are both derivable from potentials it is possible to define a total potential energy such that the total energy T + V is conserved. The term on the right is called the internal potential energy. For rigid bodies the internal potential energy will be constant. For a rigid body the internal forces do no work and the internal potential energy remains constant. Constraints • holonomic constraints: think rigid body, think f (r1, r2, r3, ..., t) = 0, think a particle constrained to move along any curve or on a given surface. • nonholonomic constraints: think walls of a gas container, think particle placed on surface of a sphere because it will eventually slide down part of the way but will fall off, not moving along the curve of the sphere. 1. rheonomous constraints: time is an explicit variable...example: bead on moving wire 2. scleronomous constraints: equations of contraint are NOT explicitly de- pendent on time...example: bead on rigid curved wire fixed in space Difficulties with constraints: 4