A Classical Approach of Newtonian Mechanics

5 CONSERVATION OF ENERGY 5.7 Motion in a general 1 - dimensional potential

k

0

The total energy of the mass shown in Fig. 42 is the sum of its kinetic and

potential energies:

E = K + U = K +

1

k x^2. (5.45)

2

Of course, E remains constant during the mass’s motion. Hence, the above ex-
pression can be rearranged to give

K = E −

1

k x^2. (5.46)

2

Since it is impossible for a kinetic energy to be negative, the above expression

suggests that |x| can never exceed the value

x =

‚

., 2 E

. (5.47)

Here, x 0 is termed the amplitude of the mass’s motion. Note that when x attains its

maximum value x 0 , or its minimum value −x 0 , the kinetic energy is momentarily

zero (i.e., K = 0 ).

5.7 Motion in a general 1 - dimensional potential

Suppose that the curve U(x) in Fig. 43 represents the potential energy of some

mass m moving in a 1-dimensional conservative force-field. For instance, U(x)

might represent the gravitational potential energy of a cyclist freewheeling in a
hilly region. Note that we have set the potential energy at infinity to zero. This is

a useful, and quite common, convention (recall that potential energy is undefined

to within an arbitrary additive constant). What can we deduce about the motion

of the mass in this potential?

Well, we know that the total energy, E—which is the sum of the kinetic energy,

K, and the potential energy, U—is a constant of the motion. Hence, we can write

K(x) = E − U(x). (5.48)

Now, we also know that a kinetic energy can never be negative, so the above

expression tells us that the motion of the mass is restricted to the region (or