Nonuniform Circular Motion
!! In all strictness, we have derived the expression for
radial acceleration for uniform circular motion
!! The derived formula(e) actually apply for nonuniform circular motion, as long as the radius of the
trajectory is constant:
1
Nonuniform Circular Motion
!! In all strictness, we have derived the expression for
radial acceleration for uniform circular motion
!! The derived
formula(e)
actually
apply
for
nonThese hold as long as r is constant,
uniform
circular
motion,
as
long
as
the
radius
of
the
regardless of what happens to vinst and !inst
trajectory is constant:
2
Linear
Angular
Displacement
Avg.
Velocity
Inst.
Velocity
3
Linear
Angular
Displacement
Avg. Velocity
Inst. Velocity
Average
Acceleration
Instantaneous
Acceleration
4
Constant Angular Acceleration
vs Constant Linear Acceleration
linear, a is constant
angular, " is constant
5
H-ITT #1: angular analogy
6
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
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]
Colloquial:!
Symmetric !
Physics term:!
Parity !
8
PHY2053, Lecture 11, Conservation of Energy!
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
9
PHY2053, Lecture 11, Conservation of Energy!
More practically speaking..
!! Key concepts:
!! Learn to recognize and exploit conserved quantities
!! Conserved quantities derived from Newton s laws
!! Solutions immediately satisfy Newton s laws
10
PHY2053, Lecture 11, Conservation of Energy!
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
!! The total energy in the Universe is unchanged by any
physical process
!! next: define change of energy (work), energy itself
11
PHY2053, Lecture 11, Conservation of Energy!
Concept of Work
!! Colloquial meaning of work: effort that
produces a result. In terms of mechanics:
!! Effort ! Force, F
!! Result ! Displacement #r
!! interested in displacement due to force
!
W = F ∆r cos(θ)
!! angular term cos(") projects force # displacement
!! SI unit: Joule [ J ]; relation to calorie: 1 cal = 4.2 J
PHY2053, Lecture 11, Conservation of Energy!
12
Work: signed scalar quantity
!! Work can be positive, negative, and zero depending
on the orientation of the force to the displacement
= 90°!
!
F!
r!
< 90°!
cos > 0!
W > 0!
F!
F!
r!
= 90°!
cos = 0!
W = 0!
PHY2053, Lecture 11, Conservation of Energy!
!
r!
> 90°!
cos < 0!
W < 0!
13
Total Work in a Closed System
!! start with total work on a particular object
W =
X
Wi =
X
i
Fi ∆r cos(θi )
i
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PHY2053, Lecture 11, Conservation of Energy!
Total Work in a Closed System
W =
X
Wi =
i
X
F~i ∆r cos ✓i = ∆r
i
X
F~i cos ✓i
i
!! Recall the definition of a closed system
!! Vector sum, has to be zero in all directions:
X
F~i = 0
i
W =
X
i
X
Wi = ∆r
F~i cos ✓i = 0
iX
i
F~i cos ✓i = 0
15
Kinetic Energy, Definition
•! consider impact of work on the velocity of an object
•! start from 1D motion, works in all three (x,
y, z)
2
2
vf,x
vi,x
ax ∆x =
−
W = Fx ∆x = max ∆x
2
2
!
2
2
2
2
vf,x
vf,x
vi,x
vi,x
W = max ∆x = m
−
=m
−m
2
2
2
2
2
v
K=m
2
W = Kf − Ki = ∆K
PHY2053, Lecture 11, Conservation of Energy!
16
Kinetic Energy, Definition
Definition of kinetic energy
2
v
K=m
2
W = Kf − Ki = ∆K
Work Energy Theorem
17
PHY2053, Lecture 11, Conservation of Energy!
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
18
PHY2053, Lecture 11, Conservation of Energy!
Mass driver notes pt 1
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PHY2053, Lecture 11, Conservation of Energy!
Mass driver notes pt 2
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PHY2053, Lecture 11, Conservation of Energy!
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: Fy = $mg
!! only y displacement (#y) matters for computing work:
W = FG,y%#y = $mg % #y
21
PHY2053, Lecture 11, Conservation of Energy!
Gravitational Potential Energy Near Earth
!! consider a vertical shot upwards, vf = 0
!! W = &K = Kf $ Ki = 0 $ 'mvf2,
!! also W = $mg % #y
!! gravity did negative work, removing kinetic energy
22
PHY2053, Lecture 11, Conservation of Energy!
Energy Conservation Law
!! where did the kinetic energy go? temporarily stored
in gravitational field
!! define potential energy &Ugrav = $Wgrav = mg % #y
!! computes how much kinetic energy could be
released if we let gravity work across #y
23
PHY2053, Lecture 11, Conservation of Energy!
Energy Conservation Law (II)
!!
!!
!!
!!
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
24
PHY2053, Lecture 11, Conservation of Energy!
Clicker Test #1: Rollercoaster
25
PHY2053, Lecture 11, Conservation of Energy!
Clicker notes:
•!
Velocity of an object moving down a frictionless slope only depends on
the height that the object started from at rest [shown in Lecture 8,
Sept 15th]!
•!
Since both objects start and end at the same height, they will have
the same velocity as they leave their tracks. This means that they will
both fly equally far. This eliminates options 4 and 5 and leaves
options 1,2 and 3!
•!
Reminder: the velocity of an object only depends on the height from
which the object started. Whichever ball is lower at a given point, is
moving faster. The curved track ball will reach the end first because
it is always below the straight track.!
26
PHY2053, Lecture 11, Conservation of Energy!
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 Wfriction < 0
27
PHY2053, Lecture 11, Conservation of Energy!
Dissipative (Non-conservative) Forces
!! friction always opposes motion, so Wfriction < 0
!! extend the law of energy conservation to account for
non-conservative forces:
(Ki + Ui) + WNC = (Kf + Uf)
28
PHY2053, Lecture 11, Conservation of Energy!
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
29
PHY2053, Lecture 11, Conservation of Energy!
Choice of Zero Point, Near Earth
!! 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)
30
PHY2053, Lecture 11, Conservation of Energy!
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?
31
PHY2053, Lecture 11, Conservation of Energy!
Example #1: Rollercoaster
h!
R!
32
PHY2053, Lecture 11, Conservation of Energy!
Rollercoaster notes pt 1
33
PHY2053, Lecture 11, Conservation of Energy!
Rollercoaster notes pt 2
34
PHY2053, Lecture 11, Conservation of Energy!
Rollercoaster notes pt 3
35
PHY2053, Lecture 11, Conservation of Energy!
Rollercoaster notes pt 4
36
PHY2053, Lecture 11, Conservation of Energy!
Rollercoaster notes pt 5
37
PHY2053, Lecture 11, Conservation of Energy!
Rollercoaster, final notes
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.
38
PHY2053, Lecture 11, Conservation of Energy!
Bowling Ball Demo
Gravitational Potential Energy, Planetary Scales
!! 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
40
PHY2053, Lecture 11, Conservation of Energy!
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?
41
PHY2053, Lecture 11, Conservation of Energy!
Comet notes, part 1:
42
PHY2053, Lecture 11, Conservation of Energy!
Comet notes, part 2:
43
PHY2053, Lecture 11, Conservation of Energy!
Comet notes, part 3:
44
PHY2053, Lecture 11, Conservation of Energy!
Comet notes, part 4:
45
PHY2053, Lecture 11, Conservation of Energy!
Next Lecture:
Hooke s Law,
Elastic Potential Energy
Power