E. Classical Mechanics
Classical mechanics was the accepted version of mechanics prior to the discovery
of relativistic mechanics and quantum mechanics. It is valid for large energies, large
masses, and speeds that are small compared with the speed of light. It is often called
Newtonian mechanics, since it was largely discovered by Isaac Newton.
E.1 Newton’s Laws of Motion
Classical mechanics is based on Newton’s three laws. The first law is thelaw of inertia:
If not acted upon by a force, a stationary object remains stationary, and a moving object
continues to move in a straight line at a constant speed.
Newton’s second law is thelaw of acceleration, which states that a force on an
object produces an acceleration inversely proportional to its mass:
Fmam
dv
dt
d^2 r
dt^2
(E-1a)
or
iFx+jFy+kFzm
(
i
d^2 x
dt^2
+j
d^2 y
dt^2
+k
d^2 z
dt^2
)
(E-1b)
wheremis the mass of the particle and wherei,j, andkare unit vectors in the direction
of thex,y, andzcoordinate axes, respectively. The accelerationais the time derivative
of the velocityvand the second time derivative of the position vector.
Newton’s third law is thelaw of action and reaction:If one object exerts a force
on a second object, the second object exerts a force on the first object that is equal in
magnitude to the first force and opposite in direction.
If the force on a particle is a known function of position, Eq. (E-1) is anequation of
motion, which determines the particle’s position and velocity for all values of the time
if the position and velocity are known for a single time. Classical mechanics is thus said
to bedeterministic. The state of a system in classical mechanics is specified by giving
the position and velocity of every particle in the system. All mechanical quantities such
as kinetic energy and potential energy have values that are determined by the values of
these coordinates and velocities, and are mechanical state functions. The kinetic energy
of a point-mass particle is a state function that depends on its velocity:
K
1
2
mv^2
1
2
m(v^2 x+v^2 y+v^2 z) (E-2)
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