conservation laws phy1012f work gregor leigh [email protected]

CONSERVATION LAWS

PHY1012

F

WORK

Gregor [email protected]

CONSERVATION LAWS WORKPHY1012F

2

WORK

Learning outcomes:At the end of this chapter you should be able to…

Extend the law of conservation of energy to include the thermal energy of isolated systems.

Calculate the work done on and by systems and apply the work-kinetic energy theorem to the solution of problems.

Distinguish between conservative and nonconservative forces.

Calculate the rate of energy transfer (power).


3

THERMAL ENERGY

An object as a whole has:Kinetic energy, K (due to movement)

Potential energy, U (due to position)

Mechanical energy, Emech

Particles within an object (i.e. atoms or molecules)

have:Kinetic energy (associated with the substance’s temperature)

Potential energy (associated with the substance’s phase)

Thermal energy, Eth


4

SYSTEM ENERGY

The sum of a system’s mechanical energy and the thermal energy of its internal particles is called the system energy, Esys.

Esys = Emech + Eth = K + U + Eth

Conversions between energy types within the system

are called energy transformations.

Energy exchanges between the system and its environment are called energy transfers.


5

ENERGY TRANSFORMATIONS

Isolated system no energy enters or leaves the system.

Transformations are indicated with arrows: e.g. K Eth .

Conversions between K and U are easily reversible, but we say that Emech is dissipated when it is

transformed into Eth since it is

extremely difficult to transform Eth back into Emech.

Friction is a common cause of the dissipation of mechanical energy.

SYSTEM

K

U

Eth

Esys

Emech + Eth


6

Esys = W

ENERGY TRANSFERS

The exchange of energy between a system and its environment by mechanical means (i.e. through the

agency of forces) is called work, W.

Energy can also be transferred by the non-mechanical process of heat. (Thermodynamics is not covered by this course.)

Work is regarded as a system asset:

work done on the system by the environ-ment increases the system’s energy: W > 0.

work done by the system on the environ-ment decreases the system’s energy: W < 0.

SYSTEM

= K + U + Eth

K

U

Eth

ENVIRONMENT

ENVIRONMENT

W < 0

Q < 0

W > 0

Q > 0work heat

work heat


7

WORK and KINETIC ENERGY

Consider a body sliding on a frictionless surface, under the action of some (possibly varying) force…

…as it moves from an initial position, si, to a final position, sf …

F

ssi , vis sf , vfs

Fs Fs

sdv dsm

ds dt

Newton II: sdvm

dt

sF ss

dvmv

ds(chain rule)

s s sF ds mv dv

F

s sF ma


8

WORK and KINETIC ENERGY

… mvs dvs = Fs ds

f f

i i

v ss s sv s

mv dv F ds

½ mvf2 – ½ mvi

2 f

i

sss

F ds

Work done by moving the object from si to

sf.

f

i

sss

F ds W F

Hence K = W

No work is done if sf = si. I.e. To do work, the

force must cause the body to undergo displacement.

Units: [N m = (kg m/s2) m = kg m2/s2 = joule, J]

Notes:


9

WORK DONE ON A SYSTEM

Work-Kinetic energy theorem:When one or more forces act on a particle as it is displaced from an initial position to a final position, the net work done on the particle by these forces causes the particle’s kinetic energy to change by K = Wnet.

f

i

s

ssK W F ds

Fs

s

Force curveThe work, W, done on a system is given by the area under a F-vs-s graph.

(cf. Impulse, J, and F-vs-t graphs.)

K p. I.e. you cannot change one without changing the other, since… 2 2

212 2 2

pmvK mvm m

displacement


10

f

i

coss

sW F ds

f

i

coss

sF ds

WORK DONE BY A CONSTANT FORCE

In the special case of a constant force…

F

ssi sf

Fs s

W Energy transfer

0° to < 90°

90°

90° to 180°

F(s)…F(s) cos Esys incr; K (and v) incr.

F(s) cos…–F(s)

0 Esys, K (and v) constant.

Esys decr; K (and v) decr.

f

i

sss

W F ds

f icosF s s cosF s

s

F

and

CONSERVATION LAWS

ˆ ˆ ˆ î i = i i cos0 = 1

WORKPHY1012F

11

THE DOT PRODUCT

The quantity F(s) cos is the product of the two

vectors, force, , and displacement, , and is more elegantly written as the dot product of the two vectors, .

F

s

ˆ ˆ ˆ î j i jx y x yF s F F s s

F s

ˆ ˆ ˆ ˆ ˆ î i i j j jx x x y y x y yF s F s F s F s F s

y

x

1

1

i

j

Note first:ˆ ˆ ˆ î j = i jcos90 0 and:

x x y yF s F s F s

I.e. the dot product is the sum of the products of the components.

ˆ ˆ= j j


12

THE DOT PRODUCT

is the angle between the two vectors.

Since it is a scalar quantity, the dot product is also known as the scalar product.

Vectors can also be multiplied using a different procedure (the cross product) to produce a vector product (q.v.).

Notes:

cosx x y yA B A B A B AB


13

Instead…

If the force varies in a simple way, we can calculate the work geometrically, by plotting and determining the area under a F-vs-s graph.

Otherwise the integral must evaluated mathematically.

WORK DONE BY A VARIABLE FORCE

If the force applied to a system varies during the course of the motion, we cannot take Fs out of the

integral…

f

i

sss

W F ds

Fnet s (N)

s (m)2 4 60

4

8

0


14

WORK DONE BY GRAVITY

y

w

s

s

dy = –cos ds

ds

Consider an object sliding down an arbitrarily-shaped frictionless surface as it moves a short distance ds.

grav cosdW w s mg ds

gravdW mg dy

f f

i igrav

s y

s ydW mg dy f img y y

Work done by gravity is thus path-independent.

Gravity is therefore a conservative force.

Notes:

Wgrav = –mgy

n


15

CONSERVATIVE FORCES

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

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

Conservative forces transform mechanical energy losslessly between the two forms, kinetic and potential.

Any conservative force has associated with it its own form of potential energy: the work done by a conservative force in moving a particle from an initial position i to a final position f, denoted Wc(if) , changes

the potential energy of the particle according to: U = –Wc(if)


16

Potential energy… Work done by…

WORK DONE BY CONSERVATIVE FORCES and POTENTIAL ENERGY

Force of gravity, :

U = –Wc

Spring force, :

Wgrav = –mgy

Wsp = –½ k(s)2

Gravitational, Ug:

Elastic, Usp:

Usp = ½ k(s)2

Ug = mgy

w

spF


17

NONCONSERVATIVE FORCES

The work done by nonconservative forces is path-

dependent. s

s

kf

Wfric = fk(s)cos180° = –kmgs

Whether the block slides directly to point A, or via point B, makes a difference to s and hence to Wfric.

B

A

All kinetic frictional forces and drag forces are nonconservative forces.

E.g. The work done by friction is


18

NONCONSERVATIVE FORCES

A nonconservative force has no associated form of potential energy. Instead, the work done by a nonconservative force increases the thermal energy, Eth, of the system – a form of energy which has no

“potential” for being reconverted to mechanical energy.

A nonconservative force is consequently known as a dissipative force. Thus: Eth = –Wdiss.

Wdiss is always negative since the force opposes motion.

Thus Eth is always positive.

Hence dissipative forces always increase the thermal

energy of a system, and never decrease it.


19

CONSERVATION OF ENERGY

(work-kinetic energy theorem)

K = Wnet

K = Wc + Wnc

K = –U + Wnc (i.e. K + U = Emech = Wnc)

K = –U + Wdiss + Wext

K = –U – Eth + Wext

K + U + Eth = Wext

Esys = Wext (energy equation of the system)

(choose system carefully to include all dissipative forces)


20

LAW OF CONSERVATION OF ENERGY

The total energy Esys = Emech + Eth of an isolated system is

a constant.

The kinetic, potential and thermal energies within the system can be transformed into each other, but their sum cannot change.

Further, the mechanical energy Emech = K + U is

conserved if the system is both isolated and non dissipative.

CONSERVATION LAWS

dWPdt

WORKPHY1012F

21

POWER

Power is the rate at which energy is transformed or transferred:

sysdEP

dt

Units: [J/s = watt, W]

Power is also the rate at which work is done:

Hence: P = Fv cos

F dsdt

dsFdt

F v


22

WORK

Learning outcomes:At the end of this chapter you should be able to…

Extend the law of conservation of energy to include the thermal energy of isolated systems.

Calculate the work done on and by systems and apply the work-kinetic energy theorem to the solution of problems.

Distinguish between conservative and nonconservative forces.

Calculate the rate of energy transfer (power).

conservation laws phy1012f work gregor leigh [email protected]

Documents

system energy

systems energy

energy types

exchange of energy

energy exchanges

position mechanical

systems mechanical energy

w energy transfers