PHYS 172: Modern Mechanics
Spring 2011
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Lecture 12 – Energy in Macroscopic System
Read 6.12 - 6.17, 7.3
Application: escape speed
Minimal condition for escape:
K +U = 0
Assume: change in kinetic energy of a planet is negligible, v<<c
Ki + U i =
2
mvesc
Mm
+ −G
=0
R
2
2
mvesc
Mm
=G
2
R
vesc =
2GM
R
Earth:
vesc = 11.2 km/s
Bound state: K
Unbound state:
+U < 0
K +U ≥ 0
See examples p. 241-243, 249, 259
Energy versus forces
Energy: predict some aspects of motion
Limits of possible
Less details
U gravitational near the Earth surface
y
Change in potential energy:
yf
∆U = ∆ −G
yi
x
Mm
Mm
Mm
= −G
− −G
r
R+h
R
∆U = GMm
( R + h) − R
R ( R + h)
∆U ≈ GMm
h
M
=m G 2 h
2
R
R
Assumption: h << R
g
∆U ≈ mgh = mg ∆y = ∆ ( mgy )
Note: motion in horizontal direction does not change U !
Path independence of potential energy
Change in potential energy does not depend on the path taken
∆U = U B − U A = −Wint
B
A
In a round trip the potential energy does not change
B
vi
vf
? At which point the gravitational potential energy is largest?
C
Clicker question
1
s
A
g
2
h
An object of mass m was pushed over the hill
of height h from point 1 to point 2 as shown.
The trajectory path length from point 1 to 2 is s.
How much work was done by the gravitational force?
A.
B.
C.
D.
E.
0J
mgh
2mgh
mgh/2
mgs
Electric potential energy
Fg = −G
U g = −G
m1m2
rˆ
r2
Fe =
analogy
1 q1q2
rˆ
4πε 0 r 2
Electric potential energy
m1m2
r
Ue =
1 q1q2
4πε 0 r
Repulsion
Attraction
See example on page 261 (6.14)
Mass and energy
Example: neutron decay
electron
antineutrino
neutron
proton
Mass of products < mass of neutron!
Missing mass is converted into kinetic energy
Example: electron and positron (antielectron) annihilation:
e − + e+ → γ + γ
2me c 2 = 2 Eγ
Eγ = 0.511 MeV
Positron tomography
All mass is converted into
electromagnetic radiation –
gamma rays
Mass and energy
Mass and energy are the same thing!
There is no “mass conservation law”!
Energy, not mass is conserved!
Mass of a multiparticle system
system
M=
Esystem = Mc 2
Esystem
c2
m1c 2 + m2 c 2 + ... + K1 + K 2 + ... + U
=
c2
Mass of a multiparticle system:
M = ( m1 + m2 + ...) +
K1 + K 2 + ... + U
c2
i>Clicker question 1
O2 molecule is oscillating:
Mass of the molecule is:
A) Larger than total mass of two noninteracting oxygen atoms
B) Smaller than total mass of two noninteracting oxygen atoms
C) The same as mass of two noninteracting oxygen atoms
M = ( m1 + m2 + ...) +
K1 + K 2 + ... + U
c2
i>Clicker question 2
O2 molecule is oscillating:
Mass of the molecule is:
A) Larger when atoms are further apart
B) Larger when atoms are closer to each other
C) Larger when atoms are moving faster
D) It does not change during oscillation
M = ( m1 + m2 + ...) +
K1 + K 2 + ... + U
c2
Two ways of thinking
1. The energy of a multiparticle system
consists of the individual particle energies
plus their pair-wise interactions
2. A system itself has energy, like a single
particle, and if the system is at rest (not
individual objects within the system!) its
energy E = Mc2, where M is mass of the
system.
Clicker
1. What happens with the mass of an object when
you heat it up?
A) It increases
B) It decreases
C)It does not change
Why don’t we notice it?
What is the change in mass when we heat up 1 kg of water from 0C to 100C?
Later we will learn that energy Q = 4.2×105 J is needed to heat it
4.2 × 105 J
Q
=
= 4.67 ×10−12 kg
∆M = 2
2
8
c
( 3 ×10 m / s )
0 C: 1.000 000 000 000 kg
100 C: 1.000 000 000 014 kg
Binding energy: nuclear reactions
1u ≡
1
m 12 = 1.660539 × 10 −27 kg
12 C
Proton: m = 1.007 276 5 u
Adding these, get
2.015 941 4 u
Neutron: m = 1.008 664 9 u
Deuteron: m = 2.013 552 8 u
Difference is 0.002 388 6 u
Binding energy, ∆mc2 =2.2MeV
(1 eV = 1×10-19 J)
Nuclear fission
slow neutron
This process generates more than 1 neutron in product, they can
initiate another reaction: chain reaction
Convert 1 pound into energy – enough to power one household for 100,000 years!