Classical MechanicsP 521, Fall 2013

Notes based on:Fetter and Walecka, Theoretical Mechanics of Particles and Continua,

Chapters 1-6

Radovan DermisekIndiana University, Bloomington

1

Basic Principlesbased on FW-1Newton’s first law:

In the primary inertial frame, every object remains at rest or in uniform motion unless acted on by a force F.

(Galileo)

Newton’s first law is valid in inertial reference frames. We usually assume that reference frames fixed on the Earth or with respect to the fixed stars are inertial frames. Any other frame that moves with constant velocity with respect to an inertial frame is also an inertial frame. Rotating and accelerating frames are not inertial frames.

2

Newton’s second law:

In the primary inertial frame, application of a force on an object alters the momentum of the object by:

if mass is constant

a=

mass is a measure of inertia

Newton’s third law:

To each action, there is an equal and opposite reaction. Thus if F is the force exerted on particle 1 by particle 2, then

and these forces act along the line separating the particles.

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3

Examples of forces:

Gravitational force:

Coulomb’s force:

1

21r

2r

1r - 2r

4

rate of change of L:

Conservation of momentum, angular momentum and energy:

Linear momentum:

0

Angular momentum:

if mass is constant(r = v)

if F = 0 then p = const.

if ! = 0 then L = const.

torque

(depends on the choice of coordinate frame)

5

Vector product (cross product):

Useful properties: In components:

6

Scalar product (dot product):

Useful properties:

7

Energy and Work:

the work done by a static force F on a particle positioned at r and moved an infinitesimal distance ds

work done in moving from point 1 to point 2:

the work done on a particle corresponds to the increase in the kinetic energy

ds = vdt

8

For conservative forces: potential (energy)(scalar function)

the work done depends only on the initial and final point(does not depend on the path taken)

W1→2 = T2 − T1combining with we get:

the total mechanical energy is conserved

alternative definitions of conservative forces:

(Arfken, 1.13)

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Gradient:

Del, or Nabla:

Divergence:

Useful properties:

Curl:

)

Stoke’s theorem:

10

total mechanical energy conserved

total mechanical energy not conserved

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12

Systems of Particlesbased on FW-2Center of Mass:

i = 1, ..., N

total mass:

13

Total force:

the total force acting on particle i

external forceon particle i

force exerted on particle i by particle j

Newton’s second law implies:

0

the center of mass R moves as if the total external force acted on the total mass concentrated at RTotal momentum:

P is constant if the external force vanishes!14

Total angular momentum:

L is constant if the external torque vanishes!

0(r = v)

= 0if force lies along ir - jr

Changes in L and P arise only from external forces!15

Working in the center of mass coordinate system:

coordinate of the ith particle measured from R

taking time derivative:

Total angular momentum:

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The total angular momentum:

the angular momentum of the center of mass relative to the fixed origin

the internal angular momentum about the center of mass

taking time derivatives:

or but more importantly:

rate of change of the internal angular momentum = the external torque about the center of mass

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The total kinetic energy:

(in the CM system)

For conservative forces:

motion of the center of mass

internal motion about the CM

assume isotropic potential,force lies along ir - jr

18

work done on the system (from configuration 1 to configuration 2):

}for 1 particle we found:

Conservation of total mechanical energy:

internal potential energy,can be also written as:external potential energy

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