PHY2053 Lecture 11
Conservation of Energy
Conservation of Energy
Kinetic Energy
Gravitational Potential Energy
Symmetries in Physics
• Symmetry - fundamental / descriptive
property of the Universe itself [“vacuum”]
• Laws of Physics are the same at any point
Colloquial:
“Symmetric”
in space [“translational invariance”]
• Conservation of Momentum [Ch 7]
• Laws of Physics are the same at any point
in time [“time invariance”]
• Conservation of Energy [today’s lecture]
PHY2053, Lecture 11, Conservation of Energy
Physics term:
“Parity”
2
More practical aspect
• there are different, mathematically equivalent ways to
formulate Newton’s laws
• all these calculations predict certain quantities will be
conserved for a “closed” system (0 net external force)
• energy, momentum, angular momentum ..
• existence of conserved quantities simplifies otherwise
complicated calculations
• Key concepts:
• learn to recognize and exploit conserved quantities
• conserved quantities derived from Newton’s laws
• solutions immediately satisfy Newton’s laws
PHY2053, Lecture 11, Conservation of Energy
3
Energy Conservation
• term “closed system” means: no net external force is
acting upon any element of the system
• The total energy of a closed system does not change
over time: total energy before = total energy after
• textbook implies that the Universe is a closed system
“The total energy in the Universe is unchanged by
any physical process”
• next: define change of energy (work), energy itself
PHY2053, Lecture 11, Conservation of Energy
4
•
•
•
•
•
Concept
of
Work
⇥
F
2,1
meaning of work:
colloquial
effort which produces a result.
analogy in terms of mechanics:
⇥ 1,2
Effort → Force, F F
Result → Displacement ∆r
interested in displacement due to force
W =F
F
2
r
θ
r
∆
r cos( )
projects force displacement
• angular term cos(θ)
X
to calorie: 1 cal = 4.2 J
⇥
• SI unit: Joule [ J ]; relation
Fi = 0
PHY2053, Lecture 11, Conservation of Energy
5
Work: signed scalar quantity
• Work can be positive, negative, and zero depending
on the orientation of the force to the displacement
F
θ
∆r
θ < 90°
cosθ > 0
W>0
θ = 90°
F
∆r
θ = 90°
cosθ = 0
W=0
PHY2053, Lecture 11, Conservation of Energy
F
θ
∆r
θ > 90°
cosθ < 0
W<0
6
F2,1
i
W =F
(3)
i
rWcos(
= F)
r cos(
)
i
r
cos(
)
(5)
Total
Work
in
a
Closed
System
F ⇥
cos( ) = 0 F⇥ = 0 X F r cos((8)
)=0
Wr
•
X
=F
i
i
F1,2
X
⇥
F
=
iW
i
X
grav
X
r•
Xi
i
i
(4)
X
0
(6)
W
=
F
r
cos(
)
X i) = 0 r
Fi r cos(
Fi cos((9)
=
i) = 0
Fi r cos(i i ) = 0
r cos(
(5)
i )
recall the m1 m2
•
F
r
cos(
)
=
0
r
i
i
U
= G
X
X
i
⇥i = 0
F
i
i
i on a particular object
start
with
total
work
X
i
W i=i F
i
i
X
⇥ 1,2
X
F
Xof
X
definition
⇥ i = 0r
F
X
i
a closed system X
⇥
(7)
Fiicos(
= 0X
F
i) = 0
Wi(10)
=
r
Fi cos( ii) = 0
(6)
X
i
i
i
W
=
i
F
r
cos(
F
cos(
)
=
0
i
i
i has to be zeroi )in=
vector
sum,
all0directions
X
Ugrav =
Fi r cos(
X
Wi =
i
i)
=0
i
m1 mi2
X
U
=
G
grav
r
F cos( ) = 0
r
Wi =
m
m
1
2
(8)
G
r
(7)
Fi cos( i ) r= 0
(9)
m
m
1
2
X
i
Ugrav = G
i
r
F
cos(
)
=
0
(8)
r
i
i
X
PHY2053, Lecture 11,
7
mConservation
m
1
2 of Energy
X
ax
WiW== Fx
W
=
F
x
=
ma
x
x
2
2
x = ma x
x(9
vf,x vi,x
2
i2
a
x
=
Kinetic
Energy,
Definition
x
x = vf,x vi,x
(12)
2
2
2
2
2
2
m m
x
1x x
2 = v 2 ax
x
=
v
v
2
a
v
f,x
f,x
i,x
consider
an
objecti,x(10
Ugrav =impact
G of work on the2velocity of
!
2
2
2
2
r
v
v
v
start from
works
in
all
three
(x,
y,
z)
vf,x 1D vmotion,
f,x2
i,x
f,
i,x
2
W
=
ma
x
=
m
=
m
v
v
2
2
x
=
(13)
x
x
f,x
i,x
v
v
f,x
i,x 2
a2x x =
2
2
2
W = Fx x = max x 2 ax 2 x =
(11
2
2
!
!
2
2
2
2
2
2
2
2
2
vf,x vi,x
v
v
v
vf,xf,x vi,x i,x vf,x
vi,x
!
2
2m (14)
m W = max x =
= m2
Km==m
m
2 m
v
v
f,x
i,x 2 (12
2 a2x x =2W
vf,x= ma
vi,x
2 2 x =2 m 2 22
=m
•
•
x
2
v 2
K = m vf,x
ax x =2
2
=K
K
2 =
2
2
v
K
=
m
2
vW
2f
i,x = K
2
Ki =
K(15)
2Work Energy Theorem
W = Kf
!
2K
PHY2053, Lecture 11, Conservation of Energy
Ki =
2
K
2
(13
8
(16)
Example #1: Mass Driver
A mass driver is a device which uses magnetic fields to
accelerate a container (mass). Predicted commercial
uses include launching people and cargo to bases on
the Moon. The common way to specify mass drivers is
to quote the kinetic energy that an object will have
when leaving the driver, if it started from rest. For a
1 MJ mass driver, compute the muzzle velocity of
a) a 0.5 kg projectile
b) a 50 kg projectile
PHY2053, Lecture 11, Conservation of Energy
9
Mass driver notes pt 1
PHY2053, Lecture 11, Conservation of Energy
10
Mass driver notes pt 2
PHY2053, Lecture 11, Conservation of Energy
11
Gravitational Potential Energy
Near Earth
• near Earth, the usual orientation of coordinate
systems is so that the positive y axis points “up”
• the force of gravity has only one component,
in the y-direction: F = −mg
• only y displacement, ∆y matters for computing work:
W = F ×∆y = −mg × ∆y
• consider a vertical shot upwards, v = 0
• W = ∆K = K − K = 0 − ½mv , also = −mg × ∆y
• gravity did negative work, “removing” kinetic energy
y
G,y
f
f
i
PHY2053, Lecture 11, Conservation of Energy
2
f
12
Energy Conservation Law
• where did the kinetic energy go? temporarily stored
in gravitational field
• define potential energy ∆U = −W = mg × ∆y
• computes how much kinetic energy could be
grav
grav
released if we let gravity work across ∆y
• work-energy theorem: W = ∆K; ∆K − W = 0
• ∆K + ∆U = 0 → ∆( K + U ) = 0
• sum of kinetic and potential energy does not change
• define E = K + U, then E is constant in time
PHY2053, Lecture 11, Conservation of Energy
13
Choice of Zero Point,
Near Earth
• Due to conservation of energy, only changes in
potential energy are really relevant for kinematics
• The absolute value of potential energy at a point in
space is arbitrary - up to an additive constant
• We have the freedom to pick a convenient point in
•
•
space and declare that the potential energy at that
point equals 0 J
All other potential energies are then computed
relative to that point, based on ∆U = U(y) − U(0)
U(y) = ∆U + U(0) = mg × ∆y + 0 = mg × (y − 0)
PHY2053, Lecture 11, Conservation of Energy
14
Example #1: Rollercoaster
A roller-coaster is barely moving as
it starts down a ramp of height h.
The first figure it encounters is a
loop of radius R. How high must the
ramp be so that the roller-coaster
never loses contact with the rails?
h
PHY2053, Lecture 11, Conservation of Energy
R
15
Rollercoaster notes pt 1
PHY2053, Lecture 11, Conservation of Energy
16
Rollercoaster notes pt 2
Comment: Given that the total height of the loop is 2R, this
is not really much taller than the loop itself. The ratio of
the height of the ramp and the height of the loop is 2.5R /
2R = 1.25 - the ramp has to be only 25% taller than the
loop for the rollercoaster to clear the highest point in the
loop and stay in contact with the rails.
PHY2053, Lecture 11, Conservation of Energy
17
More Realistic: Dissipative
(Non-conservative) Forces
• friction converts mechanical energy into heat
• heat does not “store” mechanical energy
• therefore, there is no point in defining a “heat” or
“frictional” potential energy
• friction always opposes motion, so W < 0
• extend the law of energy conservation to account for
friction
non-conservative forces:
(Ki + Ui) + WNC = (Kf + Uf)
PHY2053, Lecture 11, Conservation of Energy
18
Gravitational
Potential
Energy,
X
Wi =
Planetary
Scales
i
•
i
derivation requires math
beyond baseline calculus
Ugrav =
m1 m2
G
r
• for gravitational potential at planetary scales, there
already exists a “usual” convention:
• potential energy infinitely far away from a planet is = 0
• convention: an object with positive total energy can
“escape” a planet (will not fall back to the planet)
• allows easy computation of “escape” velocities for
objects starting from any R from the planet’s center
PHY2053, Lecture 11, Conservation of Energy
19
Example #2: Hyperbolic Comet
A comet not bound to the Sun will
only pass by the Sun once. It will trace
a hyperbolic trajectory through the
Solar system. Compute the minimum
velocity of a hyperbolic comet when it
is roughly 1 A.U. away from the Sun.
The mass of the Sun is MS = 2×1030
kg. 1 Astronomical Unit is the
distance from the Earth to the Sun,
150 million km. Does the velocity
depend on the mass of the comet?
PHY2053, Lecture 11, Conservation of Energy
20
Hyperbolic Comet notes
PHY2053, Lecture 11, Conservation of Energy
21
Next Lecture:
Hooke’s Law,
Elastic Potential Energy
Power