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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html
Lecture 13
Chapter 8
Conservation of Energy
10.23.2013Physics I
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Chapter 8
Conservative & Non-conservative forces Potential Energy Gravitational Pot. Energy Elastic Pot. Energy Conservation of Mechanical Energy
Outline
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Conservative ForcesThe work done by a conservative force in moving an object from point A to point B depends only on the positions A and B, not the path or the velocity of the object
Conservative forces: gravity, spring, electrostaticNon-conservative forces: friction, drag
The net work done by a conservative force for a round trip and returning an object to its initial position is zero
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Gravitational Potential EnergyConsider a block sliding down on a frictionless surface under the influence of gravity
x
y
d
1y gm
1K
2y
2K
)ˆ( jmggmFG
)ˆ()ˆ( jdyidxld
The work done by the gravitational force:
)]ˆ()ˆ([)ˆ(2
1
2
1
jdyidxjmgldFW GG
)( 12
2
1
yymgdymgWy
yG
KWG Work-Kinetic Energy Principle 212
1222
1 mvmv 212
1222
1 mvmv GW)( 12 yymg
Rearrange it: 1212
12
222
1 mgymvmgymv
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Conservation of Mechanical Energy!!! 1
212
12
222
1 mgymvmgymv
mgyU mgy represents a new form of energy, potential energy
Gravitational potential energy
1122 UKUK
E K UTotal Mechanical Energy
constantEConservation of Mechanical Energy Energy is transformed between kinetic and potential
gotwesoEE ,12
As the object falls, it reveals its potential energy in form of kinetic one and can do work
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
For a system, where only conservative forces do work, we have:
KW U
K2 U2 K1 U1
Relation between potential energy and work:
Work-KE Principle1221 KKUU WK
WU
U W FG.d
l
1
2In general, we define the change in potential energy associated with a conservative force F as the negative of the work done by that force.
General Potential Energy
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Potential Energy
Potential energy can only be associated with conservative forces
Energy defined as the ability to do work
Kinetic Energy: associated with energy of motion
Other types of stored energy that can do work A compressed spring An object at a height that can roll or drop
These systems have the potential to do work Call it a stored potential energy
Kinetic Energy 2
21 mvK
Only changes of potential energy important, not absolute valuesChoose a suitable reference U=0 for each problem
http://phys23p.sl.psu.edu/phys_anim/mech/ramp_n_jump.avi
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Example: Roller coasterA 1000-kg roller coaster moves from point 1 to points 2 and 3. What is the potential energy of the roller coaster at points 2 and 3 relative to point 1?What is the change in potential energy from points 2 to 3?
U1 0First, choose a reference level
U2 U1 98kJ
U2 U1 mg(y2 y1)
U3 U1 147kJ
U3 U2 (147 98)kJ 245kJ
U1 0
02 U
03 U
Subtract them:
)( 1313 yymgUU
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Example: Dropping ballAn object of mass m is dropped from a height h above the ground.Find speed of the object as it hits the ground:
F mg
vf2 vi
2 2gh
vi 0
vf 2gh
iiff mgymvmgymv 2212
21
12 mvf
2 mgh
iiff UKUK Equations of motion for constant acceleration
Energy conservation
From N. 2nd law we got this kinematic eq-n:
0
0
Thus, both approaches are equivalent
hy
Ref. level U=0
vi 0
?fv
0 h
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Elastic/Spring Potential EnergyWhat is the potential energy of a spring compressed from equilibrium by a distance x?
Fx kx
x
S ldFUxUU0
)0()(
x
kxdxU0
Uspring 12
kx2
Choose U = 0 when x = 0
WU Use a relation between potential energy and work:
2
21 kx
Potential energy of a spring
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Example: Brick/spring on a track (II)
A 2 kg mass, with an initial velocity of 5 m/s, slides down the frictionless track shown below and into a spring with spring constant k=250 N/m. How far is the spring compressed?
mx 72.
yi=2m
Ref. level U=02
212
212
21 kxmgymvmgymv ffii
spffii UUKUK
Energy conservation:
0
final
initials
miv 5
0
ii gyvkmx 2
212
So, the spring compression, x:
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Force Potential Energy
Given a conservative force as a function of position, the change in potential energy associated with this (conservative) force is:
F(x) dU(x)dx
U U(x)U(0) Fx dx0
x
Given a potential energy as a function of position, the associated conservative force is:
Force Potential Energy
Potential Energy Force
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Example: 1D Force Potential Energy
Given the potential energy:
U(x) Ax2 2Bx C
F(x) dUdx
2Ax 2B ddx
(Ax2 2Bx C)
find the force F as a function of x
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Example: Potential Energy 3D Force
In 3DF(x, y, z) U
xi U
yj U
zk
U(x, y, z) 3xy 4zUx
3y Uy
3x Uz
4
Partial Derivative: When taking derivatives with respect to one variable, treat other variables as constants
SoF(x, y, z) 3yi 3xj 4k
Potential energy is a scalar, force is a vector
ConcepTest 1 Water Slide IA) Paul
B) Kathleen
C) both the same
Paul and Kathleen start from rest at
the same time on frictionless water
slides with different shapes. At the
bottom, whose velocity is greater?
Conservation of Energy:
therefore:
Because they both start from the same height, they have the same velocity at the bottom.
fi EE 2
21 mvmgh ghv 2
Ref. level U=0
ConcepTest 2 Water Slide II
Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. Who makes it to the bottom first?
Even though they both have the same final velocity, Kathleen is at a lower height than Paul for most of her ride. Thus, she always has a larger velocity during her ride and therefore arrives earlier!
A) Paul
B) Kathleen
C) both the same
http://phys23p.sl.psu.edu/phys_anim/mech/ramped.avi Ref. level U=0
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Example: Conservation of EnergyMartin’s tighty-whiteys
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
∆y=5 m
M=40kg
k 78 N/m
kymg 0
k mgy m 5
)m/s 8.9(kg) 40( 2
(a) What is the spring constant of Martin’s tighty-whiteys?
gm
spF
N. 2nd law
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
(b) What is the potential spring energy before the cougar lets go?
y =
5 m
M=40kg
2
21 kxUiSp
θ=37°
2)37sin5(78
21
J2710
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
(c) What is Martin’s launch speed? y=
5m
M=40kg
smv 6
JkxUiSp 271021 2
θ=37°UG =0
2212
212
212
21 kxmgymvkxmgymv ffii
fSpfGfiSpiGi UUKUUK
Energy conservation:
0 0
ff mgykxm
v 2212
initial
0 0 0
0
final
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
(d) What horizontal distance does he travel?
y0=5 m
θ=37°
Vo=6 m/s
Projectile motion problem, not conservation of energy. Use kinematic equations.
y=0 (ground)
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 13
Thank youSee you on Monday