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