3Forces

IA Dynamics and Relativity

3.1 Force and potential energy in one dimension

To define the potential, consider a particle of mass

m

moving in a straight line

with position

x

(

t

). Suppose

F

=

F

(

x

), i.e. it depends on position only. We

define the potential energy as follows:

Definition

(Potential energy)

.

Given a force field

F

=

F

(

x

), we define the

potential energy to be a function V (x) such that

F = −

dV

dx

.

or

V = −

Z

F dx.

V

is defined up to a constant term. We usually pick a constant of integration

such that the potential drops to 0 at infinity.

Using the potential, we can write the equation of motion as

m¨x = −

dV

dx

.

There is a reason why we call this the potential energy. We usually consider

it to be an energy of some sort. In particular, we define the total energy of a

system to be

Definition (Total energy). The total energy of a system is given by

E = T + V,

where V is the potential energy and T =

1

2

m ˙x

2

is the kinetic energy.

If the force of a system is derived from a potential, we can show that energy

is conserved.

Proposition. Suppose the equation of a particle satisfies

m¨x = −

dV

dx

.

Then the total energy

E = T + V =

1

2

m ˙x

2

+ V (x)

is conserved, i.e.

˙

E = 0.

Proof.

dE

dt

= m ˙x¨x +

dV

dx

˙x

= ˙x

m¨x +

dV

dx

= 0

Example. Consider the harmonic oscillator, whose potential is given by

V =

1

2

kx

2

.

Then the equation of motion is

m¨x = −kx.

This is the case of, say, Hooke’s Law for a spring.

The general solution of this is

x(t) = A cos(ωt) + B sin(ωt)

with ω =

p

k/m.

A and B are arbitrary constants, and are related to the initial position and

velocity by x(0) = A, ˙x(0) = ωB.

For a general potential energy

V

(

x

), conservation of energy allows us to solve

the problem formally:

E =

1

2

m ˙x

2

+ V (x)

Since E is a constant, from this equation, we have

dx

dt

= ±

r

2

m

(E − V (x))

t − t

0

= ±

Z

dx

q

2

m

(E − V (x))

.

To find

x

(

t

), we need to do the integral and then solve for

x

. This is usually

not possible by analytical methods, but we can approximate the solution by

numerical methods.