Chapter 1, Section 1.2: Mechanics of a Many Particle System

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Sect. 1.2 Mechanics of a System of Particles

• Generalization to many (N) particle system
• Distinguish External Internal Forces.
• Newtons 2nd Law (eqtn. of motion), particle i
• ?jFji Fi(e) (dpi/dt) pi
• Fi(e) ? Total external force on the i th
  particle.
• Fji ? Total (internal) force on the i th particle
  due to the j th particle.
• Fjj 0 of course!!

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• ?jFji Fi(e) (dpi/dt) pi
  (1)
• Assumption Internal forces Fji obey Newtons 3rd
  Law Fji - Fij
• ? The Weak Law of Action and Reaction
• Original form of the 3rd Law, but is not
  satisfied by all forces!
• Sum (1) over all particles in the system
• ?i,j(?i)Fji ?iFi(e) ?i (dpi/dt)
• d(?imivi)/dt d2(?imiri)/dt2

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Newtons 2nd Law for Many Particle Systems

• Rewrite as
• d2(?imiri)/dt2 ?i Fi(e) ?i,j(?i)Fji
  (2)
• ?i Fi(e) ? total external force on system
  ? F(e)
• ?i,j(?i)Fji ? 0. By Newtons 3rd Law
• Fji - Fij ? Fji Fij 0 (cancel pairwise!)
• So, (2) becomes (ri ? position vector of mi)
• d2(?imiri)/dt2 F(e)
  (3)
• ? Only external forces enter Newtons 2nd Law to
  get the equation of motion of a many particle
  system!!

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• d2(?imiri)/dt2 F(e)
  (3)
• Modify (3) by defining R ? mass weighted average
  of position vectors ri .
• R ? (?imiri)/(?imi) ? (?imiri)/M
• M ? ?imi (total mass of
• particles in system)
• R ? Center of mass of the
• system (schematic in Figure)
• ? (3) becomes
• M(d2R/dt2) MA M(dV/dt) (dP/dt) F(e) (4)
• Just like the eqtn of motion for mass M at
  position R under the force F(e) !

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• M(d2R/dt2) F(e)
  (4)
• ? Newtons 2nd Law for a many particle
  system The 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!
• Corollary Purely internal forces (assuming they
  obey Newtons 3rd Law) have no effect on the
  motion of the Center of Mass (CM).
• Examples 1. Exploding shell Fragments travel AS
  IF the shell were still in one piece. 2. Jet
  Rocket propulsion Exhaust gases at high v are
  balanced by the forward motion of the vehicle.

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Momentum Conservation

• MR (?imiri). Consider Time derivative (const
  M)
• M(dR/dt) MV ?imi(dri)/dt ? ?imivi ? ?ipi ?
  P
• (total momentum momentum of CM)
• ? Using the definition of P, Newtons 2nd Law
  is
• (dP/dt) F(e)
  (4)
• Suppose F(e) 0 ? (dP/dt) P 0
• ? P constant (conserved)
• Conservation Theorem for the Linear Momentum
  of a System of Particles
• If the total external force, F(e), is zero, the
  total linear momentum, P, is conserved.

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Angular Momentum

• Angular momentum L of a many particle system (sum
  of angular momenta of each particle) L ? ?iri ?
  pi
• Time derivative L (dL/dt) ?idri ? pi/dt
• ?i(dri/dt) ? pi ?iri ? (dpi/dt)
• (dri/dt) ? pi vi ? (mivi) 0
• ? (dL/dt) ?iri ? (dpi/dt)
• Newtons 2nd Law (dpi/dt) Fi(e) ?j(?i) Fji
• Fi(e) ? Total external force on the i th particle
• ?j(?i) Fji ? Total internal force on the i th
  particle due to interactions with all other
  particles (j) in the system
• ? (dL/dt) ?iri ? Fi(e) ?i,j(?i) ri ? Fji

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• (dL/dt) ?iri ? Fi(e) ?i,j(?i) ri ? Fji
  (1)
• Consider the 2nd sum look at each particle pair
  (i,j). Each term ri ? Fji has a corresponding
  term rj ? Fij. Take together use Newtons 3rd
  Law
• ? ri?Fji rj?Fij ri ?Fji rj ? (-Fji)
  (ri - rj) ? Fji
• (ri - rj) ? rij vector from particle j to
  particle i. (Figure)

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• (dL/dt) ?iri ? Fi(e) (½)?i,j(?i)rij ?
  Fji (1)
• Assumption Internal forces are Central Forces
  Directed the along lines joining the particle
  pairs
• (? The Strong Law of Action and Reaction)
• ? rij Fji for each (i,j) rij ? Fji 0
  for each (i,j)!
• ? 2nd term in (1) is (½)?i,j(?i) rij ? Fji
  0

? To Prevent Double Counting!
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• ? (dL/dt) ?iri ? Fi(e)
  (2)
• Total external torque on particle i
• Ni(e) ? ri ? Fi(e)
• (2) becomes
• (dL/dt) N(e)
  (2)
• N(e) ? ?iri ? Fi(e) ?iNi(e)
• Total external torque on the system

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• (dL/dt) N(e)
  (2)
• ? Newtons 2nd Law (rotational motion) for a many
  particle system The time derivative of the total
  angular momentum is equal to the total external
  torque.
• Suppose N(e) 0 ? (dL/dt) L 0
• ? L constant (conserved)
• Conservation Theorem for the Total Angular
  Momentum of a Many Particle System
• If the total external torque, N(e), is zero,
  then (dL/dt) 0 and the total angular momentum,
  L, is conserved.

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• (dL/dt) N(e). A vector equation! Holds
  component by component. ? Angular momentum
  conservation holds component by component. For
  example, if Nz(e) 0, Lz is conserved.
• Linear Angular Momentum Conservation Laws
• Conservation of Linear Momentum holds if internal
  forces obey the Weak Law of Action and
  Reaction Only Newtons 3rd Law Fji - Fij is
  required to hold!
• Conservation of Angular Momentum holds if
  internal forces obey the Strong Law of Action
  and Reaction Newtons 3rd Law Fji - Fij
  holds, PLUS the forces must be Central Forces, so
  that rij Fji for each (i,j)!
• Valid for many common forces (gravity,
  electrostatic). Not valid for some (magnetic
  forces, etc.). See text discussion.

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Center of Mass Relative Coordinates

• More on angular momentum. Search for an analogous
  relation to what we had for linear momentum
• P M(dR/dt) MV
• Want Total momentum Momentum of CM Same as
  if entire mass of system were at CM.
• Start with total the angular momentum L ? ?iri
  ? pi
• R ? CM coordinate (origin O). For particle i
  define
• ri ? ri - R relative coordinate vector from
  CM to particle i (Figure)

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• ri ri R
• Time derivative (dri/dt) (dri/dt) (dR/dt)
  or
• vi vi V , V ? CM velocity relative to O
• vi ? velocity of particle i relative to CM.
  Also
• pi ? mivi ? momentum of particle i relative
  to O
• Put this into angular momentum
• L ?iri ? pi ?i(ri R) ? mi(vi
  V)
• Manipulation (using mivi d(miri)/dt )
• L R ? ?i(mi)V ?iri ? (mivi)
• ?i(miri) ? V R ? d?i(miri)/dt
• Note ?i(miri) defines the CM coordinate with
  respect to the CM is thus zero!! ?i(miri) ? 0
  !
• ? The last 2 terms are zero!

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• ? L R ? ?i(mi)V ?iri ? (mivi)
  (1)
• Note that ?i(mi) ? M total mass also
• mivi ? pi momentum of particle i relative
  to the CM
• ? L R?(MV) ?iri ? pi R?P ?iri?pi
  (2)
• The total angular momentum about point O the
  angular momentum of the motion of the CM the
  angular momentum of motion about the CM
• (2) ? In general, L depends on the origin O,
  through the vector R. Only if the CM is at rest
  with respect to O, will the first term in (2)
  vanish. Then only then will L be independent of
  the point of reference. Then only then will L
  angular momentum about the CM

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Work Energy

• The work done by all forces in changing the
  system from configuration 1 to configuration 2
  
• W12 ? ?i ?Fi?dsi (limits from 1 to 2)
  (1) As before Fi Fi(e) ?jFji
• ? W12 ?i ?Fi(e) ?dsi ?i,j(?i) ? Fji ?dsi
  (2)
• Work with (1) first
• Newtons 2nd Law ? Fi mi(dvi/dt). Also dsi
  vidt
• Fi?dsi mi(dvi/dt)?dsi mi(dvi/dt)?vidt
• mividvi d(½)mi(vi)2
• ? W12 ?i ? d(½)mi(vi)2 ? T2 - T1
• where T ? (½)?imi(vi)2 Total System Kinetic
  Energy

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Work-Energy Principle

• W12 T2 - T1 ?T
• The total Work done The change in kinetic
  energy
• (Work-Energy Principle or Work-Energy Theorem)
• Total Kinetic Energy T ? (½)?imi(vi)2
• Another useful form Use transformation to CM
  relative coordinates vi V vi , V ? CM
  velocity relative to O, vi ? velocity of
  particle i relative to CM.
• ? T ? (½)?imi(V vi)?(V vi)
• T (½)(?imi)V2 (½)?imi(vi)2 V??imi vi
• Last term V?d(?imi ri)/dt. From the angular
  momentum discussion ?i mi ri 0 ? The last
  term is zero!
• ? Total KE T (½)MV2 (½)?imi(vi)2

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Total KE

• T (½)MV2 (½)?i mi(vi)2
• The total Kinetic Energy of a many particle
  system is equal to the Kinetic Energy of the CM
  plus the Kinetic Energy of motion about the CM.

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Work PE

• 2 forms for work
• W12 ?i ?Fi?dsi T2 - T1 ?T (just showed!)
  (1)
• W12 ?i ?Fi(e) ?dsi ?i,j(?i) ?Fji ?dsi
  (2)
• Use (2) with Conservative Force Assumptions
• 1. External Forces ? Potential functions Vi(ri)
  exist such that (for each particle i) Fi(e) -
  ?iVi(ri)
• 2. Internal Forces ? Potential functions Vij
  exist such that (for each particle pair i,j)
  Fij - ?iVij
• 2. Strong Law of Action-Reaction ? Potential
  functions Vij(rij) are functions only of distance
  
• rij ri - rj between i j the forces lie
  along line joining them (Central Forces!) Vij
  Vij(rij)
• ? Fij - ?iVij ?jVij -Fji (ri -
  rj)f(rij)

f is a scalar ?function!
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• Conservative external forces
• ? ?i ?Fi(e) ?dsi - ?i ? ?iVi?dsi - ?i(Vi)2
  ?i(Vi)1 Or ?i ?Fi(e) ?dsi (V(e))1 -
  (V(e))2
• Where V(e) ? ?iVi total PE associated with
  external forces.
• Conservative internal forces Write (sum over
  pairs)
• ? ?i,j(?i) ?Fji?dsi (½)?i,j(?i) ?Fji?dsi
  Fij?dsj
• - (½)?i,j(?i) ??iVij?dsi
  ?jVij?dsj
• Note ?iVij - ?jVij ?ijVij (?ij ? grad
  with respect to rij)
• Also dsi - dsj drij
• ? ?i,j(?i) ?Fji?dsi - (½)?i,j(?i) ?
  ?ijVij?drij
• - (½)?i,j(?i)(Vij)2 (½)?i,j(?i)(Vij)1

do integral!!
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• Conservative internal (Central!) forces
• ?i,j(?i) ?Fji?dsi - (½)?i,j(?i)(Vij)2
  (½)?i,j(?i)(Vij)1
• or ?i,j(?i) ?Fji?dsi (V(I))1 - (V(I))2
• Where V(I) ? (½)?i,j(?i)Vij Total PE
  associated with internal forces.
• For conservative external forces conservative,
  central internal forces, it is possible to define
  a potential energy function for the system
• V ? V(e) V(I) ? ?iVi (½)?i,j(?i)Vij

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Conservation of Mechanical Energy

• For conservative external forces conservative,
  central internal forces
• The total work done in a process is
• W12 V1 - V2 - ?V
• with V ? V(e) V(I) ? ?iVi (½)?i,j(?i) Vij
• In general
• W12 T2 - T1 ?T
• Combining ? V1 - V2 T2 - T1 or ?T ?V 0
• or T1 V1 T2 V2
• or E T V constant
• E T V ? Total Mechanical Energy
• (or just Total Energy)

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Energy Conservation

• ?T ?V 0
• or T1 V1 T2 V2
• or E T V constant (conserved)
• Energy Conservation Theorem for a Many Particle
  System
• If only conservative external forces
  conservative,
• central internal forces are acting on a system,
  then
• the total mechanical energy of the system,
• E T V, is conserved.

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• Consider the potential energy
• V ? V(e) V(I) ? ?iVi (½)?i,j(?i) Vij
• 2nd term V(I) ? (½)?i,j(?i) Vij ? Internal
  Potential Energy of the System. This is generally
  non-zero might vary with time.
• Special Case Rigid Body System of particles in
  which distances rij are fixed (do not vary with
  time). (Chapters 4 5!)
• ? drij are all ? rij thus to internal forces
  Fij
• ? Fij do no work. ? V(I) constant
• Since V is arbitrary to within an additive
  constant, we can ignore V(I) for rigid bodies
  only.