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Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Lecture 15
Chapter 10
Potential EnergyConservation of Energy
Physics I
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Today we are going to discuss:
Chapter 10:
Potential Energy: Section 10.1-2 (don’t read it. Only if you have a strong desire)
Spring Potential Energy: Section 10.3 Conservation of Mechanical Energy: Section 10.4
IN THIS CHAPTER, we will add a new very important player to our energy game team (KE, work): potential energy.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Conservative Forces (definition)
The 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, springNon-conservative forces: friction
A
BCBA WWW
1
2
CFWork done by F is the same for any path
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Gravitational Potential EnergyConsider a block sliding down on a frictionless surface under the influence of gravity
x
y
sd
1y gm
1K
2y
2K
)ˆ( jmggmFG
)ˆ()ˆ( jdyidxsd
Work done by the gravitational force:
sdFW GG
2
1
dymgWy
yG
2
1
The work done by gravity depends only on coordinates of the final and initial positions, so gravitational force is conservative
)( 12 yymg
)]ˆ()ˆ([)ˆ(2
1
jdyidxjmg )ˆˆ( ij 90cosˆˆ ij 0)ˆˆ( jj 0cosˆˆ jj 1
You see there is exactly the same structure of both terms, mgy, so let’s give it a nice name and symbol
mgyU (a new form of energy)
Gravitational potential energy
)( 12 UUWG UWG
mgyUU 0Actually, in general it isReference point
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
KW U
K2 U2 K1 U1
Relation between potential energy and work
1212 )( KKUU
Only changes of potential energy important, not absolute valuesChoose a suitable reference U0=0 for each problem (like a PE origin)
Combine
Work-KE Principle
withUW KW
Conservation of Mechanical Energy!!!
So we got “a new construction” K+U, so let’s give it a nice name and symbol also
E K UTotal Mechanical Energy
constantEWhich is Conservation of Mechanical Energy
12 EE
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Energy
Potential energy can only be associated with conservative forces
Energy is 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
2
21 mvK
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
The roller-coaster car starts from rest at the top of the hill. The height of the hill is 40 m. Calculate a) the speed of the car at the bottom of
the hill;b) at what height it will have half this
speed.
Roller coasterExample
ConcepTest 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 (for any of them):
fi EE
221 mvmgh ghv 2
Ref. level U=0
ffii UKUK i
f
therefore:
because they both start from the same height (h), they have the same velocityat the bottom.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Elastic/Spring Potential Energy
What is the potential energy of a spring compressed from equilibrium by a distance x?
kxFsp
Uspring 12
kx2
Use a relation between potential energy and work:
Potential energy of a spring
Work done by a spring (from the previous class) 22
2 ifsp xxkW
Let’s combine them
22
2)( ifif xxkUU
From here you can see that the PE of a spring is
Where x is a displacement from an equilibrium of a spring
End of Class
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
A 2 kg mass, with an initial velocity of 5m/s, slides down the frictionless trackshown below and into a spring withspring constant k=250 N/m.How far is the spring compressed?
Brick/spring on a trackExample
ConcepTest 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 PhysicsPHYS.1410 Lecture 15 Danylov
vf2 vi
2 2gh
An object of mass m is dropped from a height h above the ground.Find speed of the object as it hits the ground:
vi 0
vf 2gh
iiff mgymvmgymv 2212
21
12 mvf
2 mgh
iiff UKUK
Kinematic equations 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
vf 2gh
Dropping ballExample
Now we are much more experiencedand We can apply two methods
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Thank youSee you on Monday
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