Physics 105A
Analytical Mechanics
Small Oscillations
Potential Energy in 3D
28 June 2016
Manuel Calderón de la Barca Sánchez
 The projectile will barely touch the
Over the Pipe
pipe.
 At the point at which they projectile
grazes the pipe (labeled by q):
q
the velocity will be tangential.
the magnitude of the velocity can be
very large so that the projectile clears
the pipe.
– For minimum speed, the projectile must at
least reach the point which is the mirror
reflection of “grazing point”.
Horizontal Range (at q) = Chord length
28 June 2016
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q
q
Over the Pipe: q Dependence
 Parabolic
trajectories for
various q :
 Dependence of v2
on q .
q=p/8
q=p/4
q=3p/8
q=p/2
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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.
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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
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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
Alternatively:
28 June 2016
C
F(r) = -ÑV(r )
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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
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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?
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Gravitational Potential for Shell
 Using this setup: we can show that massive spheres
behave like point masses. Sec 5.4.1
l
q
r
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P