624
14 Classical Mechanics and the Old Quantum Theory
be analyzed more easily. Let the horizontal coordinate x of the mass equal zero at its
equilibrium position and assign it to be positive if the spring is stretched and negative
if the spring is compressed. If the magnitude of x is not too large the force on the mass
due to the spring is given by Hooke’s law:
F x −kx
(14.2-13)
where k is called the force constant. Hooke’s law fails if a spring is stretched beyond
its elastic limit or if it is sufficiently compressed that its coils touch each other.
Hooke’s law is named for Robert Hooke,
1635–1703, one of Newton’s
contemporaries and rivals.
The harmonic oscillator has the following properties: (1) the x coordinate is not
limited to any finite range; (2) Hooke’s law is obeyed exactly for all values of x; (3)
the mass of the spring is negligible; (4) there is no friction.
Newton’s second law, Eq. (14.2-7), provides the equation of motion of the harmonic oscillator:
−kx m
d 2 x
dt 2
(14.2-14)
This is a differential equation. It is called linear because the dependent variable x enters
only to the first power and is called second order because its highest-order derivative is
the second derivative. The solution to a differential equation is a function that gives the
dependent variable (x in this case) as a function of the independent variable (t in this
case). There can be more than one solution function for a given differential equation.
The general solution of a differential equation is a family of functions that includes
nearly every solution of the equation.
Since both the sine and the cosine are proportional to the negative of their second
derivatives, a general solution for Eq. (14.2-14) can be written by inspection:
x(t) Asin(
k/mt) + Bcos(
k/mt)
(14.2-15)
where A and B are constants. This solution conforms to the fact that the general solution
of a linear second-order differential equation is represented by a formula that contains
two arbitrary constants. The expression for the velocity is obtained by differentiation:
v x (t)
dx
dt
k/m [Acos(
k/mt) − Bsin(
k/mt)]
(14.2-16)
The solution to Eq. (14.2-14) can be obtained in a more systematic way. A general
linear differential equation with constant coefficients can be written
0 c 0 x + c 1
dx
dt
+ c 2
d 2 x
dt 2 + · · ·
(14.2-17)
where the c’s represent constants. It is a fact that a solution to a linear differential
equation with constant coefficients can be written as
x(t) e
λt
(14.2-18)
where λ is a parameter that is determined by the coefficients. This solution is called a
trial solution because we try it in the equation. Substitution of this trial solution into
Eq. (14.2-14) gives
−ke
λt
m
d 2 e λt
dt 2 mλ
2 e
λt
(14.2-19)
14 Classical Mechanics and the Old Quantum Theory
be analyzed more easily. Let the horizontal coordinate x of the mass equal zero at its
equilibrium position and assign it to be positive if the spring is stretched and negative
if the spring is compressed. If the magnitude of x is not too large the force on the mass
due to the spring is given by Hooke’s law:
F x −kx
(14.2-13)
where k is called the force constant. Hooke’s law fails if a spring is stretched beyond
its elastic limit or if it is sufficiently compressed that its coils touch each other.
Hooke’s law is named for Robert Hooke,
1635–1703, one of Newton’s
contemporaries and rivals.
The harmonic oscillator has the following properties: (1) the x coordinate is not
limited to any finite range; (2) Hooke’s law is obeyed exactly for all values of x; (3)
the mass of the spring is negligible; (4) there is no friction.
Newton’s second law, Eq. (14.2-7), provides the equation of motion of the harmonic oscillator:
−kx m
d 2 x
dt 2
(14.2-14)
This is a differential equation. It is called linear because the dependent variable x enters
only to the first power and is called second order because its highest-order derivative is
the second derivative. The solution to a differential equation is a function that gives the
dependent variable (x in this case) as a function of the independent variable (t in this
case). There can be more than one solution function for a given differential equation.
The general solution of a differential equation is a family of functions that includes
nearly every solution of the equation.
Since both the sine and the cosine are proportional to the negative of their second
derivatives, a general solution for Eq. (14.2-14) can be written by inspection:
x(t) Asin(
k/mt) + Bcos(
k/mt)
(14.2-15)
where A and B are constants. This solution conforms to the fact that the general solution
of a linear second-order differential equation is represented by a formula that contains
two arbitrary constants. The expression for the velocity is obtained by differentiation:
v x (t)
dx
dt
k/m [Acos(
k/mt) − Bsin(
k/mt)]
(14.2-16)
The solution to Eq. (14.2-14) can be obtained in a more systematic way. A general
linear differential equation with constant coefficients can be written
0 c 0 x + c 1
dx
dt
+ c 2
d 2 x
dt 2 + · · ·
(14.2-17)
where the c’s represent constants. It is a fact that a solution to a linear differential
equation with constant coefficients can be written as
x(t) e
λt
(14.2-18)
where λ is a parameter that is determined by the coefficients. This solution is called a
trial solution because we try it in the equation. Substitution of this trial solution into
Eq. (14.2-14) gives
−ke
λt
m
d 2 e λt
dt 2 mλ
2 e
λt
(14.2-19)