Chapters 7, 8

Energy

Energy

• What is energy?

• Energy - is a fundamental, basic notion in physics

• Energy is a scalar, describing state of an object or a system

• Description of a system in ‘energy language’ is equivalent to a description in ‘force language’

• Energy approach is more general and more effective than the force approach

• Equations of motion of an object (system) can be derived from the energy equations

Scalar product of two vectors

• The result of the scalar (dot) multiplication of two vectors is a scalar

• Scalar products of unit vectors

cosABBA

ii ˆˆ 1ˆˆ jj

ji ˆˆ

1ˆˆ kk

0ˆˆ ki 0ˆˆ kj

0cos11 1

90cos11 0

Scalar product of two vectors

• The result of the scalar (dot) multiplication of two vectors is a scalar

• Scalar product via unit vectors

)ˆˆˆ)(ˆˆˆ( kBjBiBkAjAiABA zyxzyx

zzyyxx BABABABA

cosABBA

Some calculus

• In 1D case

dt

dxv

dt

dva

dt

dx

dx

dv

dx

vdv

dx

vd

2

2

dx

mvd

Fma net

2

2

2

2mvddxFnet

Some calculus

• In 1D case

• In 3D case, similar derivations yield

• K – kinetic energy

2

2mvddxFnet

2

2mvdrdFnet

Kd

2

2mvK

Kinetic energy

• K = mv2/2

• SI unit: kg*m2/s2 = J (Joule)

• Kinetic energy describes object’s ‘state of motion’

• Kinetic energy is a scalar

James Prescott Joule(1818 - 1889)

Chapter 7Problem 31

A 3.00-kg object has a velocity of (6.00^i – 2.00^j) m/s. (a) What is its kinetic energy at this moment? (b) What is the net work done on the object if its velocity changes to (800^i + 4.00^j) m/s?

Work–kinetic energy theorem

• Wnet – work (net)

• Work is a scalar

• Work is equal to the change in kinetic energy, i.e. work is required to produce a change in kinetic energy

• Work is done on the object by a force

dKmv

drdFnet

2

2

f

i

r

r netif rdFKK

netW

Work: graphical representation

• 1D case: Graphically - work is the area under the curve F(x)

f

i

x

x xdxFW

Net work vs. net force

• We can consider a system, with several forces acting on it

• Each force acting on the system, considered separately, produces its own work

• Since

f

i

r

r kk rdFW

) ( sumvectorFFk

knet

) ( sumscalarWWk

knet

Work done by a constant force

• If a force is constant

• If the displacement and the constant force are not parallel

f

i

r

rrdFW

f

i

r

rrdF

rF

coscos FdrFrFW

Work done by a spring force

• Hooke’s law in 1D

• From the work–kinetic energy theorem

kxFs

f

i

x

x ss dxFW f

i

x

xkxdx

22

22fi

kxkx

Work done by the gravitational force

• Gravity force is ~ constant near the surface of the Earth

• If the displacement is vertically up

• In this case the gravity force does a negative work (against the direction of motion)

cosmgdWg

180cosmgdWgmgd

Lifting an object

• We apply a force F to lift an object

• Force F does a positive work Wa

• The net work done

• If in the initial and final states the object is at rest, then the net work done is zero, and the work done by the force F is

mgdWW ga

gaifnet WWKKKW

Power

• Average power

• Instantaneous power – the rate of doing work

• SI unit: J/s = kg*m2/s3 = W (Watt)

t

WPavg

dt

dWP

James Watt(1736-1819)

Power of a constant force

• In the case of a constant force

dt

dWP

dt

rFd )(

dt

rdF

vF

cosFvP

Chapter 8Problem 32

A 650-kg elevator starts from rest. It moves upward for 3.00 s with constant acceleration until it reaches its cruising speed of 1.75 in/s. (a) What is the average power of the elevator motor during this time interval? (b) How does this power compare with the motor power when the elevator moves at its cruising speed?

Conservative forces

• The net work done by a conservative force on a particle moving around any closed path is zero

• The net work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle

02,1, baab WW

2,1, baab WW

2,2, baab WW

2,1, abab WW

Conservative forces: examples

• Gravity force

• Spring force

0 downup mghmgh

022

22

leftright kxkx

Potential energy

• For conservative forces we introduce a definition of potential energy U

• The change in potential energy of an object is being defined as being equal to the negative of the work done by conservative forces on the object

• Potential energy is associated with the arrangement of the system subject to conservative forces

WU

Potential energy

• For 1D case

• A conservative force is associated with a potential energy

• There is a freedom in defining a potential energy: adding or subtracting a constant does not change the force

• In 3D

if UUU

CdxxFxU )()(dx

xdUxF

)()(

kz

zyxUj

y

zyxUi

x

zyxUzyxF ˆ),,(ˆ),,(ˆ),,(),,(

f

i

x

xdxxFW )(

Chapter 7Problem 44

A single conservative force acting on a particle varies F = (– Ax + Bx2) ^i N, where A and B are constants and x is in meters. (a) Calculate the potential energy function U(x) associated with this force, taking U = 0 at x = 0. (b) Find the change in potential energy and the change in kinetic energy of the system as the particle moves from x = 2.00 m to x = 3.00 m.

Gravitational potential energy

• For an upward direction the y axis

f

i

y

ydymgyU )()( ymgmgymgy if

mgyyU g )(

Elastic potential energy

• For a spring obeying the Hooke’s law

f

i

x

xdxkxxU )()(

22

22

if kxkx

2)(

2kxxU s

Internal energy

• The energy associated with an object’s temperature is called its internal energy, Eint

• In this example, the friction does work and increases the internal energy of the surface

Conservation of mechanical energy

• Mechanical energy of an object is

• When a conservative force does work on the object

• In an isolated system, where only conservative forces cause energy changes, the kinetic and potential energies can change, but the mechanical energy cannot change

UKEmec

WK WU UK )( ifif UUKK

iiff UKUK imecfmec EE ,,

Work done by an external force

• Work is transferred to or from the system by means of an external force acting on that system

• The total energy of a system can change only by amounts of energy that are transferred to or from the system

• Power of energy transfer, average and intantaneous

intEUKW

t

EPavg

dt

dEP

Conservation of mechanical

energy: pendulum

Potential energy curve

dx

xdUxF

)()(

Potential energy curve: equilibrium points

Unstable equilibrium

Neutral equilibrium

Stable equilibrium

Chapter 8Problem 55

A 10.0-kg block is released from point A. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between block and the rough surface between B and C.

Answers to the even-numbered problems

Chapter 7

Problem 2:

(a) 3.28 × 10−2 J (b) - 3.28 × 10−2 J

Answers to the even-numbered problems

Chapter 7

Problem 10:

16.0

Answers to the even-numbered problems

Chapter 7

Problem 46:

(7−9x2y)ˆi−3x3ˆj

Answers to the even-numbered problems

Chapter 8

Problem 14:

(a) 0.791 m/s (b) 0.531 m/s

Answers to the even-numbered problems

Chapter 8

Problem 28:

8.01 W

Answers to the even-numbered problems

Chapter 8

Problem 34:

194 m

Answers to the even-numbered problems

Chapter 8

Problem 50:

(a) 0.588 J (b) 0.588 J(c) 2.42 m/s(d) UC = 0.392 J, KC = 0.196 J