Chapter 5: Energy

Energy Energy is present in a variety of forms: mechanical, chemical, electromagnetic, nuclear, mass, etc. Energy can be transformed from one from to another.

The total amount of energy in the Universe never changes.

If a collection of objects can exchange energy with each other but not with the rest of the Universe (an isolated system), the total energy of the system is constant. If one form of energy in an isolated system decreases, another form of energy must increase. In this chapter, we focus on mechanical energy: kinetic energy and potential energy.

Work The work W done on an object by a constant force F when the object is displaced by x by the force:

xFW SI unit: joule (J) = newton-meter (N m) = kg m2/s2

• Work is a scalar quantity.

• If the force exerted on an object is not in the same direction as the displacement:

xFxFW

)cos(

component ofthe force alongthe direction ofthe displacement

dot product orinner product

Work If an object is displaced vertical to the direction of a force exerted, then no work is done.

)90(0)cos( xFW

If an object is displaced in opposite direction to that of an exerted force, the work done by the force is negative (if F<Fg).

)180(

)cos(

xF

xFW

Work Work and dissipative forces

• The friction force between two objects in contact and in relative to each other always dissipate energy in complex ways.

• Friction is a complex process caused by numerous microscopic interactions over the entire area of the surfaces in contact.

• The dissipated energy above is converted to heat and other forms of energy.

• Frictional work is extremely important: without it Eskimos can’t pull sled, cars can’t move, etc.

Work Examples

• Example 5.1: Sledding through the Yukon

m=50.0 kgF= 1.20x102 Nx=5.00 m

(a) How much work is done if =0?

(b) How much work is done if =30o?

J 1000.6

)m 00.5)(J 1020.1(2

2

xFW

J 1020.5

)m 00.5(30cos)J 1020.1(

)cos(

2

2

xFW

Work Examples

• Example 5.2: Sledding through the Yukon (with friction)

m=50.0 kgF= 1.20x102 Nx=5.00 m

(a) How much work is done if =0?

(b) How much work is done if =30o?

fk=0.200

J1010.100

)J1090.4(J 1000.6

J1090.4

0

2

22

2

gnfricappnet

k

kkfric

y

WWWWW

xmg

xnxfW

mgnmgnF

J0.9000)J1030.4(J 1020.5

J1030.4)sin(

0

22

2

gnfricappnet

appkkkfric

y

WWWWW

xFmgxnxfW

mgnmgnF

Kinetic Energy Kinetic energy (energy associated with motion)

• Consider an object of mass m moving to the right under action of a constant net force Fnet directed to the right.

xmaxFW netnet )((constant acceleration)

xavv 220

2

2

20

2 vvxa

KEKEKEW

mvmv

vvmW

ifnet

net

20

2

20

2

2

1

2

1

2Define the kinetic energy KE as:

2

2

1mvKE SI unit: J

work-energy theorem

Kinetic Energy An example

• Example 5.3: Collision analysis

m=1.00x103 kgvi = 35.0 m/s -> 0

=8.00x103 N(a) The minimum necessary stopping distance?

m 6.762

10

2

1

2

1 222

x

mvxfmvmvW ikifnet

(b) If x=30.0 m what is the speed at impact?

m/s 3.27/m 7452

2

1

2

1

2222

22

fkif

ifkfricnet

vsxfm

vv

mvmvxfWW

Kinetic Energy Conservative and non-conservative forces

• Two kinds of forces: conservative and non-conservative forces

• Conservative forces : gravity, electric force, spring force, etc. A force is conservative if the work it does moving an object between two points is the same no matter what path is taken. It can be derived from “potential energy”.

• Non-conservative forces : friction, air drag, propulsive force, etc. In general dissipative – it tends to randomly disperse the energy of bodies on which it acts. The dispersal of energy often takes the form of heat or sound. The work done by a non-conservative force depends on what path of an object that it acts on is taken. It cannot be derived from “potential energy”.

• Work-energy theorem in terms of works by conservative and non- conservative force

KEWW cnc

Gravitational Potential Energy Gravitational work and potential energy

• Gravity is a conservative force and can be derived from a potential energy.

Work done by gravity on the book:

ymg

yymg

yymgyFW

if

fig

)(

0cos)(cos

KEymgmvKE

KE

ygvygvv

f

i

2

220

2

2

1

0

22

KEWWW gncnet )( ifnc yymgKEW

Gravitational Potential Energy Gravitational work and potential energy

• Gravity is a conservative force and can be derived from a potential energy.

)( ifnc yymgKEW • Let’s define the gravitational potential energy of a system consisting of an object of mass m located near the surface of Earth and Earth as:

mgyPE y : the vertical position of the mass to a reference point ( often at y=0 )g : the acceleration of gravitySI unit: J

PEKEWnc

)( if

if

yymg

PEPEPE

where

Gravitational Potential Energy Reference levels for gravitational potential energy

• As far as the gravitational potential is concerned, the important quantity is not y (vertical coordinate) but the difference y between two positions.

• You are free to choose a reference point at any level (but usually at y=0).

yi

yf

Gravitational Potential Energy Gravity and the conservation of mechanical energy

• When a physical quantity is conserved the numeric value of the quantity remains the same throughout the physical process.

• When there is no non-conservative force involved,

0 PEKEWnc

ffii PEKEPEKE • Define the total mechanical energy as: PEKEE • The total mechanical energy is conserved.

fi EE ffii mgymvmgymv 22

2

1

2

1

• In general, in any isolated system of objects interacting only conservative forces, the total mechanical energy of the system remains the same at all times.

Gravitational Potential Energy Examples

• Example 5.5: Platform diver(a) Find the diver’s speed at y=5.00 m.

ffii PEKEPEKE

ffii mgymvmgymv 22

2

1

2

1

ffi gyvgy 2

2

10

m/s 90.9)(2 fif yygv

(b) Find the diver’s speed at y=0.0 m.

02

10 2 fi mvmgy

m/s 0.142 if gyv

Gravitational Potential Energy Examples

• Example 5.8: Hit the ski slopes(a) Find the skier’s speed at the bottom (B).

ffii PEKEPEKE

ffii mgymvmgymv 22

2

1

2

1

ffi gyvgy 2

2

10

m/s 8.19)(2 fif yygv

(b) Find the distance traveled on the horizontal rough surface.

222

2

1

2

1

2

1BkBCknet mvmgdmvmvKEdfW

m 2.952

2

g

vd

k

B

210.0kf0kf

Spring Potential Energy Spring and Hooke’s law

• Force exerted by a spring Fs

Fs

x>0

kxFs

If x > 0, Fs <0If x < 0, Fs >0 Fs to the right

Fs to the left

Hooke’s law

• The spring always exerts its force in a direction opposite the displacement of its end and tries to restore the attached object to its original position.

Restoring force

k : a constant of proportionality called spring constant. SI unit : N/m

Spring Potential Energy Potential due to a spring• The spring Fs is associated with elastic potential energy.

-Fs

xxi+1

-Fi

width = xxi-1/2x

xi

xi+1/2x

Between xi -1/2x and xi+1/2x the workexerted by the spring is approximately: )/(,, NxFxFW isisi Between x=0 and x, the total work exertedby the spring is approximately:

2,

1

1 ,

1 ,

2

1

)(lim

lim

lim

kxxF

area

xF

WW

Ns

N

i iN

N

i isN

N

i isNs

-Ws,i= areai

In general when the spring is stretchedfrom xi to xf, the work done by the springis:

22

2

1

2

1ifs kxkxW

Spring Potential Energy Potential due to a spring (cont’d)• The energy-work theorem including a spring and gravity

gifnc PEKElxkxW

22

2

1

2

1

fsgisg PEPEKEPEPEKE )()(

Extended form of conservation of mechanical energy

)()()( sisfgigfifnc PEPEPEPEKEKEW

2

2

1kxPEs elastic potential energy

Spring Potential Energy Examples• Example 5.9: A horizontal spring

m=5.00 kgk=4.00x102 N/mxi=0.0500 m

(a) Find the speed at x=0 without friction.

0,02

1

2

1

2

1

2

1 2222

fi

ffii

xv

kxmvkxmv

m/s 447.0

22

iffi xm

kvvx

m

k

(b) Find the speed at x=xi/2.

m

kxv

m

kx ff

i

22

2

m/s 387.0)( 22 fif xxm

kv

k=0

Spring Potential Energy Examples• Example 5.9: A horizontal spring (cont’d)

m=5.00 kgk=4.00x102 N/mxi=0.0500 m

(c) Find the speed at x=0 with friction

m/s 230.0

22

ikif gxxm

kv

2222

2

1

2

1

2

1

2

1ififfric kxkxmvmvW

22

2

1

2

1ifik kxmvnx

ikif nxkxmv 22

2

1

2

1

k= 0.150

Spring Potential Energy Examples• Example 5.10 : Circus acrobat

m=50.0 kgh =2.00 mk = 8.00 x 103 N/m

What is the max. compressionof the spring d?

fsg

isg

PEPEKE

PEPEKE

)(

)(

2

2

1000)(0 kddhmg

0m 0.245-m) 123.0( 22 dd

m 560.0d

Spring Potential Energy Examples• Example 5.11 : A block projected up a frictionless incline

(a)Find the max. distance d the block travels up the incline.

m=0.500 kgxi=10.0 cmk=625 N/m=30.0o

22

22

2

1

2

12

1

2

1

fff

iii

kxmgymv

kxmgymv

m 28.1

sin

2/sin

2

1 22

mg

kxdmgdmghkx i

i

(b) Find the velocity at hafl height h/2.

ghvxm

khmgmvkx fifi

2222

2

1

2

1

2

1m/s 50.22 ghx

m

kv if

Spring Potential Energy Systems and energy conservation

• Work-energy theorem KEWW cnc

• Consider changes in potential

if

iiff

ififnc

EE

PEKEPEKE

PEPEKEKEPEKEW

)()(

)()(

The work done on a system by all non-conservative forces isequal to the change in mechanical energy of the system.

If the mechanical energy is changing, it has to be going somewhere.The energy either leaves the system and goes into the surroundingenvironment, or stays in the system and is converted into non-mechanical form(s).

Systems and Energy Conservation Forms of energy

• Forms of energy stored kinetic, potential, internal energy

• Forms of energy transfer between a non-isolated system and its environment

Mechanical work : transfers energy to a system by displacing it with a force. Heat : transfers energy through microscopic collisions between atoms or molecules. Mechanical waves : transfers energy by creating a disturbance that propagates through a medium (air etc.). Electrical transmission : transfers energy through electric currents. Electromagnetic radiation :

transfers energy in the form of electromagnetic waves such as light, microwaves, and radio waves.

Systems and Energy Conservation Energy conservation• Principle of energy conservation:

• The principle of conservation of energy is not only true in physics but also in other fields such as biology, chemistry, etc.

Energy cannot be created or destroyed, only transferred fromone form to another.

Power Power• The rate at which energy is transferred is important in the design and use of practical devices such as electrical appliances and engines.

• If an external force is applied to an object and if the work done by this force is W in time interval t, then the average power delivered to the object during this interval is the work done divided by the time interval:

vFt

xFt

WP

SI unit : watt (W) = J/s = 1 kg m2/s3

W=Ft

FvP More general definition

U.S. customary unit : 1 hp = 550 ft lb/s = 746 W1 kWh = (103 W)(3600 s) = 3.60 x 105 J

Power Examples• Example 5.12 : Power delivered by an elevator

What is the min. power to lift the elevator with the max. load?

M=1.00x103 kgm=8.00x102 kgf =4.00x103 Nv = 3.00 m/s

0

gMfT

amF

0 MgfT

N 1016.2 4 MgfT

hp 86.9 kW 64.8

W1048.6 4

FvP

Power Examples• Example 5.14 : Speedboat power

How much power would a 1.00x103 kg speed boat need to go from rest to 20.0 m/s in 5.00 s, assuming the water exerts a constant drag force of magnitude fd=5.00x102 N and the acceleration is constant?

hp 60.3 W1050.4

J 1025.22

1

J 1050.2

m 0.502

m/s 00.4

2

12

1

2

1

4

52

4

22

2

2

22

t

WP

xfmvW

xfW

xavv

aatvvatv

mvxfWWW

mvmvKEW

engine

dfengine

dfric

if

fif

fdenginedragengine

ifnet

Power Energy and power in a vertical jump• Center of mass (CM)

• Stationary jump

The point in an object at which all the may be considered to beconcentrated.

Two phases:(1) Extension, (2) free flight

ffii KEPEKEPE

g

vHmgHmv CM

CM 22

1 22

h=0.40 m depth of croucht=0.25 s time for extensionm=68 kg

m 52.0

m/s 2.3/22

H

thvvCM

W104.1

J 105.32

1

3

22

t

KEP

mvKE CM