3Forces
IA Dynamics and Relativity
3.3 Central forces
While in theory the potential can take any form it likes, most of the time, our
system has spherical symmetry. In this case, the potential depends only on the
distance from the origin.
Definition
(Central force)
.
A central force is a force with a potential
V
(
r
) that
depends only on the distance from the origin,
r
=
|r|
. Note that a central force
can be either attractive or repulsive.
When dealing with central forces, the following formula is often helpful:
Proposition. ∇r =
ˆ
r.
Intuitively, this is because the direction in which
r
increases most rapidly is
r, and the rate of increase is clearly 1. This can also be proved algebraically:
Proof. We know that
r
2
= x
2
1
+ x
2
2
+ x
2
3
.
Then
2r
∂r
∂x
i
= 2x
i
.
So
∂r
∂x
i
=
x
i
r
= (
ˆ
r)
i
.
Proposition. Let F = −∇V (r) be a central force. Then
F = −∇V = −
dV
dr
ˆ
r,
where
ˆ
r
=
r/r
is the unit vector in the radial direction pointing away from the
origin.
Proof. Using the proof above,
(∇V )
i
=
∂V
∂x
i
=
dV
dr
∂r
∂x
i
=
dV
dr
(
ˆ
r)
i
Since central forces have spherical symmetry, they give rise to an additional
conserved quantity called angular momentum.
Definition (Angular momentum). The angular momentum of a particle is
L = r × p = mr ×
˙
r.
Proposition. Angular momentum is conserved by a central force.
Proof.
dL
dt
= m
˙
r ×
˙
r + mr ×
¨
r = 0 + r × F = 0.
where the last equality comes from the fact that
F
is parallel to
r
for a central
force.
In general, for a non-central force, the rate of change of angular momentum
is the torque.
Definition
(Torque)
.
The torque
G
of a particle is the rate of change of angular
momentum.
G =
dL
dt
= r × F.
Note that
L
and
G
depends on the choice of origin. For a central force, only
the angular momentum about the center of the force is conserved.