Work Energy Theorem Variable Mass_phy Edu 2001

7/28/2019 Work Energy Theorem Variable Mass_phy Edu 2001

1/5

The work-energy theorem: the effect of varying mass

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2001 Phys. Educ. 36 61

(http://iopscience.iop.org/0031-9120/36/1/311)

Download details:

IP Address: 129.174.55.245

The article was downloaded on 12/09/2011 at 08:10

Please note that terms and conditions apply.

View the table of contents for this issue, or go to thejournal homepage for more

ome Search Collections Journals About Contact us My IOPscience
http://iopscience.iop.org/page/termshttp://iopscience.iop.org/0031-9120/36/1http://iopscience.iop.org/0031-9120http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/contacthttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/journalshttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/searchhttp://iopscience.iop.org/http://iopscience.iop.org/0031-9120http://iopscience.iop.org/0031-9120/36/1http://iopscience.iop.org/page/terms


2/5

FEATURES

The workenergy theorem:

the effect of varying mass

Ronald Newburgh

Extension School, Harvard University, Cambridge, MA 02138, USA

AbstractThis paper examines the workenergy theorem of classical physics andapplies it to two situations with non-constant mass. The first is that of arocket burning fuel. The second is that of a proton in a particle acceleratorreaching relativistic speeds. The analysis leads to a reformulation of theworkenergy relation in terms of momentum and velocity rather than forceand displacement. This allows graphical determinations of kinetic energythat are both simple to make and clear even to beginning students. The useof graphical integration obviates the necessity of using calculus for thenon-constant mass situations. An unexpected result is the invariance of thisreformulation even for the relativistic case.

Introduction

A student recently asked a question that prompted

me to re-examine the workenergy theorem andits relation to kinetic energy. His question wasquite simple. What is the meaning of a plot ofvelocity versus momentum for a body since anarea in momentumvelocity space is an energy?The question is one with several far-reachingimplications. The first of these leads to a re-examination of the workenergy theorem andallows a graphical interpretation of the kineticenergy. The graphical interpretation offers afresh look at kinetic energy that has considerablepedagogic merit, as judged from student response.

This result alone would justify the analysis.

However, the graphical technique has shown itselfto be more general and powerful, allowing, as itdoes, treatment of time-varying mass. Graphicalintegration allows one to determine net workor change in kinetic energy quite simply. Thepaper examines both the rocket problem with non-constant mass and particles moving at relativisticvelocities. This approach makes these two topicsfar more accessible to beginning students than domore conventional techniques.

Perhaps even more important is the result

that a simple equation in elementary mechanics,

namely the one relating work, momentum and

velocity, is equally valid in relativistic mechanics.Let me stress that this does not mean that there

are two equations that are covariant but rather

that there is one equation that is the same in

both relativistic and classical mechanics. This

was a completely unexpected consequence of the

analysis.

Net work and momentum

Let us begin with the definition of the net work

done on a body undergoing a displacement. For

simplicity we shall consider a net force acting

parallel to the displacement. The infinitesimal net

work, dWnet , is given as

dWnet = Fnet dx (1)

where Fnet is the net force and dx the infinitesimal

displacement. The total net work for a body

going from x1 to x2 is obtained from integrating

0031-9120/01/010061+04$30.00 2001 IOP Publishing Ltd P H Y S I C S E D U C A T I O N 61


3/5

R Newburgh

equation (1) or

Wnet =

x2x1

Fnet dx. (2)

We can plot this force versus displacement andproduce figure 1. The net work done on the objectis the area under the curve. Note that an area in

the forcedisplacement plane has units of newtons metres, or joules.

Rewriting equation (1) leads to another viewof the net work. Using Newtons second law, we

write

dW= (dp/dt)dx = dp(dx/dt) (3)

= (dp)v

or

dW/dp = v.Integrating equation (3) gives the total work as

W=

v dp (3a)

in which p is momentum, t time and v velocity.

This is a less common but equally valid form of thework equation. We shall now apply this form ofthe equation to several quite different situations.

Work and constant mass

For a body whose mass is constant the momentumis always linear with velocity, since

p = mv. (4)

In other words, the momentum has the samefunctional dependence on time as does the

velocity. Therefore plotting velocity versusmomentum gives the linear graph of figure 2. Theplot applies no matter how the force varies with

time, as long as the mass is constant. An areain the momentumvelocity plane has the units(newtons seconds)(metres/seconds) or joules.

Evaluating thearea forthe range from zero velocity

to the maximum velocity, we find the net work tobe

Wnet =12vmaxpmax =

12m(vmax)

2. (5)

This last term is the well-known expression for the

kinetic energy. More generally we can write

Wnet = KE. (5a)

The net work therefore equals the change in thekinetic energy of the body during its displacement.

Figure 1. Plot of an arbitrary force versusdisplacement. The area under the curve from x1 to x2represents the change in kinetic energy over thatinterval.

Figure 2. Plot of velocity versus momentum for abody of constant mass. Since the mass is constant, themomentum is a linear function of velocity. The area

under the graph represents the net work or the changein kinetic energy as the body accelerates from zerovelocity to a maximum value.

The rocket with varying mass

Consider a rocket accelerated vertically. Its engine

hasa continuous exhaust fired straight out from the

rocket tail. The exhaust velocity of the gas with

respect to the rocket is u. The fuel burns at the

constant rate such that

dm/dt= . (6)

The gravitational force is mg. For simplicity

we shall take g to be constant and neglect air

resistance. This is the only external force acting

on the system. Therefore we may write the net

force as

Fnet = dp/dt= m dv/dt+ u dm/dt

= m dv/dtu = mg. (7)

62 P H Y S I C S E D U C A T I O N


4/5


5/5

R Newburgh

Figure 4. Plot of velocity expressed as a fraction oflight velocity versus the momentum divided by the rest

mass. The graph represents the change in relativistickinetic energy as the proton accelerates.

The relativistic kinetic energy, T, equals the total

energy minus the rest energy or

T = E m0c2. (15)

Differentiating equation (15) we find

dT/dp = dE/dp

=d

dp[c(p2 + m0c

2)1/2 m0c2]

= pc/(p2 + m0c2)1/2

= pc2/c(p2 + m0c2)1/2

= pc2/E = v. (16)

Equation (16) is identical with equation (3) except

for the change in the definition of kinetic energy.

The area under the graph is

T =

v dp. (17)

Rosser [1] has an especially good treatment of this

material.

Discussion

Figures 2, 3 and 4 are plots of velocity versus

momentum and, as such, represent the change in

kinetic energy according to equations (3a) and

(17). Though they do represent the change in

kinetic energy, this change does not always equal

the net work.

We have treated three cases, two classical and

one relativistic. The mass is constant for the first

case (figure 2), and the velocity is linear withmomentum. Here the net work done on the object

equals its change in kinetic energy.

The second case (figure 3) treats a rocket with

non-constant, continuously varying mass. The

change in mass is a consequence of the continuous

mass loss from the fuel combustion. The change

in kinetic energy does not equal the net work, since

some of the net work is doneon the lost mass. The

kinetic energy at any instant is that of the rocket

plus the amount of fuel on board, not that of the

rocket plus the original fuel.

The relativistic case (figure 4) is like the

first one, in that no mass is lost. However,because of the relativistic mass increase with

velocity, the momentum increases more rapidly

than if the momentum were linear with velocity.

Moreover, all the work done on the particle by the

accelerator equals the increase of kinetic energy

defined relativistically. Thus, once again, net work

equals the change in kinetic energy.

Perhaps the most striking fact of this analysis

is the universality of equations (3), (16) and

(17). The only difference between the relativistic

and classical cases is the generalization of the

definition of kinetic energy. As I implied in theintroduction, this generalization is a consequence

of the invariance of the equation relating work,

momentum and velocity in both classical and

relativistic physics.

Acknowledgments

I wish to thank William Boone, a student in my

class at the Harvard Extension School, who first

raised the question of the meaning of the pv plot.

This paper is an attempt to answer his question.

Received 17 April 2000, in final form 15 August 2000

PII: S0031-9120(01)13415-5

Reference

[1] Rosser W G V 1964 Introduction to the Theory ofRelativity 1st edn (London: Butterworths)pp 1845

64 P H Y S I C S E D U C A T I O N

Work Energy Theorem Variable Mass_phy Edu 2001

Documents

Transcript of Work Energy Theorem Variable Mass_phy Edu 2001