Physics 105A
Energy conservation
 Conservation laws
“Conservation” : notion of a quantity
that does not change with time.
Restricts the possible final motions
of a system.
– Write down enough conservation laws
until only one motion is allowed.
Energy, momentum, angular
momentum conservation.
Start from F=ma and derive them.
– Later, derive them differently.
Why use them? They can greatly
simplify our life and understanding.
 Energy : can neither be created,
nor destroyed, only transformed.
28 June 2016
MCBS
Energy
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
Work-Energy Theorem (1-D)
1 2 1 2
mv2 - mv1 = - (V(x2 ) -V (x1 )) =
2
2
x2
ò F(x)dx º Work
x1 ®x2
x1
 The change in a particle’s kinetic energy between points
x1 and x2 equals the work done on the particle between x1
and x2.
If force points in the same direction as motion:
– F(x) and dx have same sign
– W x1→x2 > 0 : Positive work
– v22>v12 : velocity/kinetic energy increases
If force points in the opposite direction as motion:
– F(x) and dx have opposite sign
– W x1→x2 < 0 : Negative work
– v22<v12 : velocity/kinetic energy decreases
28 June 2016
MCBS
Potential Energy Plots
 Using the relation between
kinetic, potential and total
energy:
1 2
mv +V(x) = E
2
 Once we pick a reference
point, we can plot V(x) vs. x
 We can also plot E.
E>V: Allowed region.
E<V: Forbidden region
28 June 2016
MCBS
Energy in a Bungee Jump
 A certain physicist of mass m wants to use a bungee cord
with spring constant k. The bungee cord has unstretched
length L. How far down will the physicist fall, assuming
initial speed is zero and a constant g field? Neglect losses
due to friction and air resistance.
Sigh, No
bungees in
1687…
28 June 2016
MCBS
Gravitational Potential Energy
GMm
 Newton’s Law of Universal Gravitation F  
2
r
 Consider two point masses. What is the potential energy
of the system measured at separation r?
m
28 June 2016
M
r
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
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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
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MCBS
P