Physics 105A
Analytical Mechanics
Work, Potential and Internal
Energy
Small Oscillations
Potential Energy in 3D
28 June 2016
Manuel Calderón de la Barca Sánchez
Work or Potential Energy?
 Question: When should one use work, and when should
one use Potential Energy.
Work always works.
For conservative forces, both work. Use either (but don’t use
both!)
 What about systems with various parts?
Work done on a system by external forces equals the change
in energy of the system
– Overall Kinetic Energy
Wexternal
– Internal Potential Energy
– Internal Kinetic Energy (e.g. heat)
28 June 2016
MCBS
= DK + DV + DKinternal
Transforming Energy: A Cool Idea
Don’t cha wish your boyfriend
was smart, like me?
Don’t cha wish your boyfriend
was a geek, like me?
Consider a car braking without skidding.
Friction from ground on tires causes car to slow down (F=ma).
However, ground doesn’t move!
Force always acts over zero distance:
Wexternal=0
Total energy of car doesn’t change: DK = -DK internal
K goes down, Kint goes up.
Kinetic energy goes to heat the brake pads and discs.
Energy is lost. Can’t be converted back to overall kinetic energy of car.
What if we could convert K into some form of internal U?
d
1 2
K  U ,   
, U  C
dt
2
28 June 2016
MCBS
Small oscillations near a minimum
 Taylor expansion of V(x):
V (x) = V (x0 )+V '(x0 )(x - x0 )
1
+ V ''(x0 )(x - x0 )2
2
1
+ V '''(x0 )(x - x0 )3 + ...
3!
28 June 2016

Near minimum:
(x-x0) is a small quantity.

At minimum x0:
V’(x0) = 0

MCBS
To leading order:
1
V(x) » V ''(x0 )(x - x0 )2
2
Small oscillations example
 Find the frequency of small
oscillations for
V(x) = A/x2 – B/x
Where A, B > 0.
28 June 2016
MCBS
ì
dv x
ï Fx = m
dx
ï
ïï
dv y
F(r ) = ma = í Fy = m
dy
ï
ï
dv z
ï Fz = m
dz
ïî
Forces in 3-D
dv x
dx
=m
vx
dt
dx
dv y
dy
=m
vy
dt
dy
dv z
dz
=m
vz
dt
dz
 To obtain Work: multiply by dx, dy, dz, add and integrate.
28 June 2016
MCBS
Conservative forces in 3-D
 Given a force F(r), a necessary and sufficient condition for
the potential,
V (r) = - ò F(r ') × dr '
r
r0
to be well defined (i.e. to be path independent) is that
Ñ ´ F(r ) = 0
Use Stoke’s Theorem:
ò (Ñ ´ F ) × dA = ò F(r ) × dr
S
28 June 2016
C
MCBS
C
S
Central Forces
 Definition:
Direction of Force points radially
– Towards or away from the origin
– F(r ) = F(r )r̂
Magnitude of Force depends on distance from
origin
– F(r ) = F(r)r̂
– Where r = x 2 + y2 + z 2
 Important result: All central forces are
conservative!
Proof: Show
28 June 2016
Ñ´ F = 0
MCBS
Gravity
 Newton’s Universal Law of Gravitation
Force on a point mass m due to another
point mass M located at the origin, is given
by
GMm
F = - 2 r̂
r
– Central force
– Attractive:
 Force on m points from m towards M.
– Distance from m to M is r.
– G = 6.67 x 10-11 m3/(kg s2)
Question: What if we don’t have point
masses?
28 June 2016
MCBS
Gravitational Potential for Shell
 Using this setup: we can show that massive spheres
behave like point masses. Sec 5.4.1
l
q
r
28 June 2016
MCBS
P