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