14.2 Classical Mechanics
621
14.2
Classical Mechanics
According to classical mechanics, the state of a system is specified by giving the
position and velocity of every particle in the system. Consider a single particle without
any internal structure so that it cannot rotate or vibrate. If it can move in three dimensions we can specify its position by the three Cartesian coordinates x, y, and z. These
three coordinates are equivalent to a position vector that reaches from the origin of
coordinates to the location of the particle. Any quantity that has direction as well as
magnitude is a vector. We denote a vector by a letter in boldface type and we denote this
position vector by r. It can also be denoted by a letter with an arrow above it, as in − → r ,
by a letter with a wavy underscore, as in . The Cartesian coordinates x, y, and z are
called the Cartesian components of the position vector r. A vector can also be denoted
by listing its three Cartesian components inside parentheses, as in (x, y, z). Any quantity that has both magnitude and direction is a vector. A quantity that has no particular
direction but can be positive, negative, or zero, is called a scalar. Appendix B contains
an elementary introduction to vectors, and additional information can be found in the
references of note number 1.
The velocity of the particle is the time derivative of its position vector and is a vector
v with the Cartesian components ν x , ν y , and ν z :
ν x
dx
dt
ν y
dy
dt
ν z
dz
dt
(14.2-1a)
These three equations can be represented by a single vector equation
v
dr
dt
(14.2-1b)
The speed of a particle is the magnitude of the velocity, given by
ν |v| |
ν 2
x + ν 2
y + ν 2
z
(14.2-2)
which is a three-dimensional version of the theorem of Pythagoras. The kinetic energy
of the particle is defined by
K
1
2 mv
2
1
2 mv
2 (v
2
x + v
2
y + v
2
z )
(14.2-3)
where m is the mass of the particle. The kinetic energy is an example of a function of
the variables that specify the state of the particle. A function of these variables is called
a state function.
Units of Measurement
In using a formula such as that of Eq. (14.2-3), one must use consistent units or risk
getting the wrong answer. The set of units that chemists and physicists now use is the
international system of units, or SI units. The letters SI stand for Système Internationale,
the French name for the set of units. In this system there are seven base units. The unit
of length is the meter (m). The unit of mass is the kilogram (kg). The unit of time is
the second (s). The unit of temperature is the kelvin (K). The unit of electric current
is the ampere (A). The unit of luminous intensity is the candela (cd). The unit for the
amount of a substance is the mole (mol).
621
14.2
Classical Mechanics
According to classical mechanics, the state of a system is specified by giving the
position and velocity of every particle in the system. Consider a single particle without
any internal structure so that it cannot rotate or vibrate. If it can move in three dimensions we can specify its position by the three Cartesian coordinates x, y, and z. These
three coordinates are equivalent to a position vector that reaches from the origin of
coordinates to the location of the particle. Any quantity that has direction as well as
magnitude is a vector. We denote a vector by a letter in boldface type and we denote this
position vector by r. It can also be denoted by a letter with an arrow above it, as in − → r ,
by a letter with a wavy underscore, as in . The Cartesian coordinates x, y, and z are
called the Cartesian components of the position vector r. A vector can also be denoted
by listing its three Cartesian components inside parentheses, as in (x, y, z). Any quantity that has both magnitude and direction is a vector. A quantity that has no particular
direction but can be positive, negative, or zero, is called a scalar. Appendix B contains
an elementary introduction to vectors, and additional information can be found in the
references of note number 1.
The velocity of the particle is the time derivative of its position vector and is a vector
v with the Cartesian components ν x , ν y , and ν z :
ν x
dx
dt
ν y
dy
dt
ν z
dz
dt
(14.2-1a)
These three equations can be represented by a single vector equation
v
dr
dt
(14.2-1b)
The speed of a particle is the magnitude of the velocity, given by
ν |v| |
ν 2
x + ν 2
y + ν 2
z
(14.2-2)
which is a three-dimensional version of the theorem of Pythagoras. The kinetic energy
of the particle is defined by
K
1
2 mv
2
1
2 mv
2 (v
2
x + v
2
y + v
2
z )
(14.2-3)
where m is the mass of the particle. The kinetic energy is an example of a function of
the variables that specify the state of the particle. A function of these variables is called
a state function.
Units of Measurement
In using a formula such as that of Eq. (14.2-3), one must use consistent units or risk
getting the wrong answer. The set of units that chemists and physicists now use is the
international system of units, or SI units. The letters SI stand for Système Internationale,
the French name for the set of units. In this system there are seven base units. The unit
of length is the meter (m). The unit of mass is the kilogram (kg). The unit of time is
the second (s). The unit of temperature is the kelvin (K). The unit of electric current
is the ampere (A). The unit of luminous intensity is the candela (cd). The unit for the
amount of a substance is the mole (mol).