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Evaluation of mechanical energy in different frames of reference
Applied Energy 45 (1993) 335-345
r
Evaluation of Mechanical Energy in Different Frames of Reference H. Barrow" & D. G. C l a y t o n b a Department of Mechanical Engineering,Universityof Liverpool, UK, L69 3BX b 70 Collisdene Road, Orrell, Wigan, UK, WN5 8RL
A BS TRA CT Invoking the universality of the law of conservation of energy, it is shown that the quantifications of work and mechanical energies are dependent on the frame of reference with respect to which these are measured By means of simple isothermal examples in applied mechanics (which avoids the complication of thermal energy considerations), the veracity of the usual energy equations is demonstrated when different frames of references are used for: (1) an accelerating mass, (2) connected masses and (3) a cylinder rolling freely down an incline.
INTRODUCTION There is a widespread awareness that 'distance' and 'velocity' are dependent on the frame of reference with respect to which they are measured. However, there still remains some misunderstanding on how mechanical work is measured in different frames, possibly leading in turn to an incorrect formulation of the energy balance of a system. Interest in the rigour of clearly defining energy transfers continues, and this is reflected in papers such as Refs 1 and 2, where the emphasis is on the interpretation of work in mechanics and in thermodynamics. Furthermore, the problem of quantifying work in dry friction has attracted considerable attention; very good accounts have been given by Bridgrnan, 3 as early as 1941, and more recently by Gibbings. 4 The evaluation of work in fluid mechanics also has been explained in detail by Barrow and Pope; 5 this study is particularly relevant to control-volume analysis for a continuum. 335 Applied Energy 0306-2619/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain
H. Barrow, D. G. Clayton
336
In the present paper, the intention of which is to clarify work and energy evaluation, a number of simple mechanics examples have been chosen to demonstrate frame dependence. These include two examples in linear particle dynamics, and a rigid body in rotation down an inclined plane to show that the results are completely general. However, a brief review of the fundamental principles and concepts will be presented first.
THEORETICAL CONSIDERATIONS Newton's second law of motion is so well known that it suffices here to state the relationship between force P, mass m and acceleration a as P
(1)
= ma
The frame of reference in which this applies may be described as the Newtonian frame or stationary frame ('stationary' refers to a fixed point on the earth's surface). The validity of eqn (1) in frames of reference with motion relative to the so-called stationary frame also needs con-sideration. As a consequence of the Galilean transformation, for other frames in translation with uniform velocity, the second law applies equally well and the generalisation is described as the Newtonian principle of relativity. That is, if eqn (1) applies in a stationary frame, it will apply equally in all frames of reference moving in translation with uniform velocity. For frames which have acceleration relative to the stationary frame, eqn (1) requires modification by the introduction of a fictitious force, for, if am is the acceleration of the moving frame, then the acceleration of m in the moving frame is (a - am), and therefore P
-
mam
= m(a
-
am)
(2)
for compatibility with eqn (1). A much more detailed explanation of the considerations leading to eqn (2) has been given by Symon. 6 This matter is best appreciated by the elementary example shown in Fig. 1. The values of the 'force' and 'acceleration' in the law of motion in the two cases are as indicated in the diagrams. It needs to be realised that the 'real' force P itself is unchanged; that is, the 'real' force is absolute. This is best understood by observing an instrument which records the force P in the stationary frame. If the observer now moves with the frame as in (b), he or she will observe the same reading on the instrument. A further important point which must be made here is concerned with the kinetic energy of the mass m. This is clearly defined in terms of the velocity in the chosen frame of reference. The expression for kinetic energy [(1A)mass(velocity) 2]
Mechanical energy in different frames of reference
fixed
frame
mass,m
337
= 5 real force P acceleration : 2
:
I0
(a)
o
moving a.ccn,
frame a,,,
=
0
effective
force
acceleration
= 2
-
10
- 5
1
x
-
1
=
I
(b)
Fig. 1.
Linear motion in (a) fixed and (b) moving frames.
is obtained from a development of the fundamental equation of motion. Having been quantified in terms of the velocity in one particular frame, it must employ the velocity in any other frame where appropriate. Furthermore, kinetic energy is a scalar and advantage will be taken of this later. The main concern of the present paper is work transfer and energy in mechanics with particular reference to moving frames. An appropriate starting point is the traditional definition of work, viz. dW:
F . dS
(3)
where d W is the work done on a system, F is the force exerted on the matter in the system by the environment, and dS is the distance moved by the point of application of F. (The dot product is employed when the force and displacement are general vectors.) It must also be added that dS must be the displacement in the particular frame. As pointed out by Reynolds, 7 if the frame is attached to the matter, then there is no energy transfer as work in that case. This requires some elaboration, as reference to Fig. 2 will reveal. Let us consider first the system in motion in a particular frame. If the system is distorting, as may well be the case, the frame would need to be attached to the matter at the point of application of the force. Clearly, if there were other forces acting on the system's
i moving o
frame
t~I
G*/ "~ )
X
Fig. 2.
Work in a distorting system in a moving frame of reference.
338
H. Barrow, D. G. Clayton
boundary, some work interaction would take place, as in such a case there would be relative motion between the points of application of the forces even in a moving frame. (The distance moved by the centre of mass, G, of course is not necessarily the same as that moved by A.) Another interesting case, in connection with the work concept in mechanics, is that of ideal rolling down an inclined plane. Figure 3 shows this familiar problem. A cylinder or sphere rolls down an inclined plane without any slipping at the interface or point of contact. The interesting point is that although the observer in the stationary frame sees the force P in motion no work is done on (or by) the rolling body. This at first seems to be an anomaly, but the problem may be explained in two ways. First, the motion to and from the point of contact is perpendicular to the inclined plane if there is no distortion. Accordingly, the work is zero even though the force P moves in space; of course, P features in the momentum equation should this alternative method of solution be employed. Another more satisfying explanation of this is afforded using the model shown in Fig. 3(b). At any instant, the force P is acting at the boundary of the system on a particular element of matter within the system. As rolling proceeds, this element is replaced by the adjacent element, so that the rolling may be considered as a series of discrete steps in each of which the work is zero. So, although the force P is observed as moving in space, it does not move attached to material at the boundary of the system. It therefore seems that the definition of work given above requires some refinement with regard to 'the point of application'. (In real rolling, there is some work interaction in the region of the contact, as dry friction is ultimately the result of the motion of many elementary forces acting on elements of the system at the interface. This phenomenon requires a separate treatment and is not pursued any further here.) Attention is now drawn to those examples in applied mechanics which have been chosen to demonstrate how work and energy are evaluated in the most general case when the frame of reference is in motion relative to the Earth.
cylinder or sphere Stationary \frame f '
Fig. 3.
/
//
(a)
Ideal rolling down an inclined plane, (a) general model, (b) model of contact point.
Mechanical energy in different frames of reference
339
EXAMPLES
Linear motion of a particle An obvious starting point is the case of a particle acted on by a constant force. Four frames of reference have been chosen: (a) (b) (c) (d)
the Newtonian or stationary frame; a uniformly accelerating frame; a constant velocity frame; a frame attached to the particle.
Of interest are the velocities and distances moved in these frames (in a specified time), and the kinetic energy change and work done. The results are most conveniently presented in tabular form, and Table 1 lists the various quantities pertaining to unit mass. In all cases, the change in kinetic energy equals the work done in the frame, calculated as the effective force multiplied by the distance moved. The effective force is the real force less the fictitious force where appropriate. The important point to be emphasised is that, in general, work done depends upon the frame of reference. This important conclusion is reinforced by the results for the very special case (d), where the frame moves with the particle itself. Clearly, there is no change of mechanical energy and no work transfer; an observer with the frame sited on the particle would not be aware of any change in the motion of the particle. Although he or she observes a real force, the observer cannot detect any displacement. Finally, it should be pointed out that the data for the simple case shown in Fig. 1 are consistent with the expressions listed in Table l, cases (a) and (b). This may be verified by substitution.
Connected particles A more interesting example is that of two particles connected by an inextensible cord, as shown in Table 2. One 'mass' is in motion on a smooth horizontal table; the cord passes over a frictionless pulley and is connected to the second 'mass' which is in vertical motion. Again, the results are listed in tabular form; only those for the Newtonian frame and constant horizontal velocity frame are shown. It is a straightforward matter to obtain the corresponding results for the most general case of the horizontal accelerating frame. This case, however, requires caution, as, in an accelerating frame, the second mass sustains not only the vertical force of its weight, but also a horizontal fictitious force associated
x frame
force
(½)af2t 2
(½) a2t2
Final kinetic energy of unit mass
Change in kinetic energy of unit mass
am
am
(at - am)[-Umt + (½)(af - am)t 2]
[-um + (af - am)t]2~2
(½)(--Um) 2
(af -- am)[--Umt + (l/2)(af -- am)t 2]
ar-
- u m + (af - am)t -Umt+(l/2)(af - am)t 2
af-
--Um
Uniform acceleration, a m
af[-Umt+ (1~2)aft2]
(-urn + aft)2~2
(1/2)(-- Um)2
af[--Umt + (½)aft 2]
0
0
0
0
0
0
--Um + (l/2)aft2
a~
0 0
0
Frame attached to mass
- u m + aft
a~
--Um
Zero acceleration
Moving Frame (initial velocity, urn)
For the fixed frame and the frame moving with uniform velocity, the acceleration and the force of unit mass are the same according to Newton's principle of relativity (see Ref. 6).
0
(]/2)a2t 2
af"
aft ('/z)af t2
a~
0
Fixed frame
Initial kinetic energy of unit mass
Work done
(Effective) force unit mass
Distance moved
Final velocity of mass
Acceleration of mass
Initial velocity of mass
ftn
unit mass
TABLE 1 Linear M o t i o n of a Particle
L~
341
M e c h a n i c a l e n e r g y in different f r a m e s o f reference
TABLE 2 Connected Particles
( Newtonian )
Fixed frame
Moving frame (constant velocity urn)
Initial velocity of m
0
-- Um
Initial velocity of M
0
-- Um
Mg/(M + m)
Mg/(M + m)
ITIaSS m
frame
mass M
Acceleration of m and M Tension force P
Mmg/(M + m)
Mmg/(M + m)
Final velocity of m
Mgt/(M + m) Mgt/(M + m)
- u m + Mgt/(M + m) {(-urn) 2 + [Mgt/(M + m)]2] 1/2
MZg2tZ/[2(M + m)]
P . s m - P . s + Mgs
Final velocity of M Work done Initial kinetic energy of m
0
(V2)m(-um) 2
Initial kinetic energy of M
('A)M~ -urn) 2 (V2)m[-u~ + Mgt/(M + m)] 2
Final kinetic energy of m
0 MZmg2t2/[2(M + m) 2]
Final kinetic energy of M
M392t2/[2(M ~- m) 2]
M{(-Um)-' + [Mgt/(M + m)]2}/2
Change in total kinetic energy
M2g2t2/[2(M + m)]
(-umMmgt)/(M + m) + M2g2t2/(2(M + m)t
Distance moved by m (= Sm)
MgtZ/[2(M + m)]
- u ~ t + Mgt2/(2(M + m))
Vertical distance moved by M (=- s)
MgtZ/[2(M + m)]
Mgt2/[2(M + m)}
with the acceleration of the frame in that direction. (For the moving frame with constant horizontal velocity, there are no force effects associated with the second mass in this direction.) Despite the somewhat lengthy algebra, it may be shown that the work done is also different in magnitude, but again equal to the change of kinetic energy. There are now three components of work done (and three for kinetic energy change), corresponding to the motion of the mass m in the horizontal, and the motions of the mass M in the vertical and the horizontal directions. Of course, both work and kinetic energy are scalar quantities, and the components may be added algebraically to obtain the total energy transfer and energy change. The choice of the direction of motion of the frame of reference is arbitrary, and although the horizontal has been selected for present purposes, the calculation may be effected with, say, a vertically accelerating frame. In that case, two components of work (and energy) would be associated with the mass m, and one with the mass M. This example then is a further demonstration that in the work-energy principle in mechanics, the work and energies are calculated with respect to the frame of reference. Unlike 'real force', work and energy are not absolute, and are frame dependent.
342
H. Barrow, D. G. Clayton
Ideal rolling down an inclined plane The examples chosen so far refer to linear motion of bodies and the appropriate dynamics pertaining to those situations. The principles involved must be general and, as an illustration of this, a simple example of combined translation and rotation will now be considered. Reference was made above to ideal rolling with regard to the definition of work. This example also serves well as an illustration of the dependence of work on the frame of reference. Following the practice adopted for the previous examples of tabulating the results for various frames of reference, Table 3 has been compiled for the ideal rolling of a cylinder with a stationary frame and a frame moving in translation with the axis of the cylinder. The solution for this problem in the stationary frame does not require any elaboration, as it is a standard mechanics problem frequently used to demonstrate the work-energy principle. The results for the moving frame, however, deserve comment. Both linear velocities and accelerations are zero in this case, leaving rotational kinetic energy as the only component of energy to be considered. The point now is that the rotational dynamics may TABLE 3 Ideal R o l l i n g D o w n an Inclined Plane
sin
mass M, unit radius
Fixed frame (Newtonian)
114ovingframe (in translation attached to the axis of the cylinder)
0 2g(sin 0)/3 2g(sin O)t/3 2g(sin O)t/3 g(sin 0)t2/3
0 g2(sin2 0)t2/3 g2(sin2 0)t2/3
0 0 0 2g(sin O)t/3 0 g(sin 0)t2/3 g(sin 0)/3 0 g2(sin2 0)t2/9 gZ(sin20)t"/9
(linear and rotational)
(rotational)
1rame
Initial velocity Linear acceleration Final linear velocity Final angular velocity Linear distance (in frame) Angular distance Force F of unit mass Initial kinetic energy of unit mass Final kinetic energy of unit mass Change in kinetic energy of unit mass Work done/unit mass (effective force × distance in the frame, or effective torque × angular distance)
g(sin 0)t2/3 g(sin 0)/3
g(sin O) x g(sin O)fl/3 g(sin 0)/3 x g(sin 0)t2/3
Mechanical energy in different frames of reference
343
be calculated independently of the linear ones. As there is no rotational acceleration of the moving frame, the 'real' force P is to be used in the torque in the rotational dynamics of the body. Of course, the force P now does work to produce the rotational kinetic energy, as there is motion relative to the frame. Clearly, should the frame both move and rotate with the rolling body, all the pertinent quantities would be zero as for the simple case of a flame attached to the mass in Table 1. The most general case of a linearly and rotationally accelerating frame (comparable with that listed in column 3 of Table 1) is academic in the extreme and has not been pursued here. However, its investigation would involve the principles employed in the connected-particle example referred to above, with, of course, the additional rotational equations of motion allowing for rotational acceleration.
DISCUSSION A N D CONCLUSIONS The principal intention at the outset of the investigation reported here was to ascertain that mechanical work and hence energy change are dependent on the frame of reference in which the measurements are made. Mechanical work is an energy interaction on a system, involving (in general) changes in both thermal and mechanical energies of the system. By considering an insulated system, the complexities of quantifying changes in the thermal-energy component may be avoided and the net mechanical-work transfer then equals the change of total mechanical energy (i.e. the sum of the potential and kinetic energies). This simple 'work-energy principle' may be used to advantage to demonstrate qualitatively the dependence of work on the frame of reference. Clearly, the potential and kinetic energies of a system are themselves frame dependent, as position and velocity are measured in the frame, and so the total mechanical energy is specified uniquely. Now the 'work-energy principle' is considered as a universal physical law as a consequence of relativity considerations. Accordingly, the energy interaction, or so-called work, must also be dependent on the frame of reference. Although the above reasoning may be made without reference to a specific situation or event, it is satisfying to test such reasoning quantitatively. This has been done in the present work by examining a number of simple dynamics problems. The problems chosen conform to the conditions pertaining to the simple 'work-energy principle', i.e. thermal effects are absent. Furthermore, these problems have already been introduced at an early stage in one's academic training, and so will be familiar to all readers. The three examples chosen are the dynamics of (1) a single
344
H. Barrow, D. G. Clayton
particle, (2) a system of connected particles and (3) ideal rolling of a body down an incline. Various frames of reference (both inertial and non-inertial) have been considered, and example (3) has been selected to introduce rotational aspects of the problem. In all cases, the kinematics have been solved first, and then the total energy change calculated from the point of view of the observer in the particular frame. For the determination of the mechanical work transfer, the product F . s has been evaluated, but, of course, the force F (or forces where appropriate) is that according to whether the frame is inertial or non-inertial. In the latter case, the force is the effective force, or the real force less the fictitious or inertial force, in order that Newton's law shall hold in that frame (see eqn (2)). Of course, the displacement s is that measured in the chosen frame for compatibility with the procedure for evaluating the kinematics of the problem. It has been shown conclusively, that, if the calculations are made in this manner, the 'work-energy principle' is satisfied irrespective of the frame, as it should be. As to the relative magnitudes of the work in each of the cases considered, that may be obtained from the tables listing the results. In fact, in some cases it is possible to have zero work and, of course, zero energy change. Finally, although this investigation has shown the dependence of work transfer on the frame of reference in a particular area of study, the philosophical implications are completely general. This generalisation is reflected in the following quotation, which is to be found in The Mechanical Universe by Frautschi et al.: s 'From the beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer (moving relative to the earth), everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest' (Albert Einstein, Autobiographical Notes (1949)).
REFERENCES 1. Roberts, J. W., Aspects of work in mechanics. Int. J. Mech. Eng. Educ., 16(1) (1986) 1-16. 2. Lewins, J. D., Letter to the Editor. Int. J. Mech. Eng. Educ., 17(2) (1986) 131-5. 3. Bridgman, P. W., The Nature of Thermodynamics. Harper Torchbook, London, 1961. 4. Gibbings, J. C., Thermomechanics. Pergamon Press, Oxford, 1970. 5. Barrow, H. & Pope, C. W., Clarification of the evaluation of friction work transfer in control volume analyses. Int. J. Mech. Eng. Educ., 19(1) (1991).
Mechanical energy in different frames of reference
345
6. Symon, K. R., Mechanics. Addison Wesley, Reading, Massachusetts, USA, 1973. 7. Reynolds, W. C., Thermodynamics. McGraw-Hill, New York, 1968. 8. Frautschi, S. C., Olenick, R. P., Apostol, T. M. & Goodstein, D. L., The Mechanical Universe, Mechanics and Heat, Advanced edn. Cambridge University Press, London, 1986.
r
Evaluation of Mechanical Energy in Different Frames of Reference H. Barrow" & D. G. C l a y t o n b a Department of Mechanical Engineering,Universityof Liverpool, UK, L69 3BX b 70 Collisdene Road, Orrell, Wigan, UK, WN5 8RL
A BS TRA CT Invoking the universality of the law of conservation of energy, it is shown that the quantifications of work and mechanical energies are dependent on the frame of reference with respect to which these are measured By means of simple isothermal examples in applied mechanics (which avoids the complication of thermal energy considerations), the veracity of the usual energy equations is demonstrated when different frames of references are used for: (1) an accelerating mass, (2) connected masses and (3) a cylinder rolling freely down an incline.
INTRODUCTION There is a widespread awareness that 'distance' and 'velocity' are dependent on the frame of reference with respect to which they are measured. However, there still remains some misunderstanding on how mechanical work is measured in different frames, possibly leading in turn to an incorrect formulation of the energy balance of a system. Interest in the rigour of clearly defining energy transfers continues, and this is reflected in papers such as Refs 1 and 2, where the emphasis is on the interpretation of work in mechanics and in thermodynamics. Furthermore, the problem of quantifying work in dry friction has attracted considerable attention; very good accounts have been given by Bridgrnan, 3 as early as 1941, and more recently by Gibbings. 4 The evaluation of work in fluid mechanics also has been explained in detail by Barrow and Pope; 5 this study is particularly relevant to control-volume analysis for a continuum. 335 Applied Energy 0306-2619/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain
H. Barrow, D. G. Clayton
336
In the present paper, the intention of which is to clarify work and energy evaluation, a number of simple mechanics examples have been chosen to demonstrate frame dependence. These include two examples in linear particle dynamics, and a rigid body in rotation down an inclined plane to show that the results are completely general. However, a brief review of the fundamental principles and concepts will be presented first.
THEORETICAL CONSIDERATIONS Newton's second law of motion is so well known that it suffices here to state the relationship between force P, mass m and acceleration a as P
(1)
= ma
The frame of reference in which this applies may be described as the Newtonian frame or stationary frame ('stationary' refers to a fixed point on the earth's surface). The validity of eqn (1) in frames of reference with motion relative to the so-called stationary frame also needs con-sideration. As a consequence of the Galilean transformation, for other frames in translation with uniform velocity, the second law applies equally well and the generalisation is described as the Newtonian principle of relativity. That is, if eqn (1) applies in a stationary frame, it will apply equally in all frames of reference moving in translation with uniform velocity. For frames which have acceleration relative to the stationary frame, eqn (1) requires modification by the introduction of a fictitious force, for, if am is the acceleration of the moving frame, then the acceleration of m in the moving frame is (a - am), and therefore P
-
mam
= m(a
-
am)
(2)
for compatibility with eqn (1). A much more detailed explanation of the considerations leading to eqn (2) has been given by Symon. 6 This matter is best appreciated by the elementary example shown in Fig. 1. The values of the 'force' and 'acceleration' in the law of motion in the two cases are as indicated in the diagrams. It needs to be realised that the 'real' force P itself is unchanged; that is, the 'real' force is absolute. This is best understood by observing an instrument which records the force P in the stationary frame. If the observer now moves with the frame as in (b), he or she will observe the same reading on the instrument. A further important point which must be made here is concerned with the kinetic energy of the mass m. This is clearly defined in terms of the velocity in the chosen frame of reference. The expression for kinetic energy [(1A)mass(velocity) 2]
Mechanical energy in different frames of reference
fixed
frame
mass,m
337
= 5 real force P acceleration : 2
:
I0
(a)
o
moving a.ccn,
frame a,,,
=
0
effective
force
acceleration
= 2
-
10
- 5
1
x
-
1
=
I
(b)
Fig. 1.
Linear motion in (a) fixed and (b) moving frames.
is obtained from a development of the fundamental equation of motion. Having been quantified in terms of the velocity in one particular frame, it must employ the velocity in any other frame where appropriate. Furthermore, kinetic energy is a scalar and advantage will be taken of this later. The main concern of the present paper is work transfer and energy in mechanics with particular reference to moving frames. An appropriate starting point is the traditional definition of work, viz. dW:
F . dS
(3)
where d W is the work done on a system, F is the force exerted on the matter in the system by the environment, and dS is the distance moved by the point of application of F. (The dot product is employed when the force and displacement are general vectors.) It must also be added that dS must be the displacement in the particular frame. As pointed out by Reynolds, 7 if the frame is attached to the matter, then there is no energy transfer as work in that case. This requires some elaboration, as reference to Fig. 2 will reveal. Let us consider first the system in motion in a particular frame. If the system is distorting, as may well be the case, the frame would need to be attached to the matter at the point of application of the force. Clearly, if there were other forces acting on the system's
i moving o
frame
t~I
G*/ "~ )
X
Fig. 2.
Work in a distorting system in a moving frame of reference.
338
H. Barrow, D. G. Clayton
boundary, some work interaction would take place, as in such a case there would be relative motion between the points of application of the forces even in a moving frame. (The distance moved by the centre of mass, G, of course is not necessarily the same as that moved by A.) Another interesting case, in connection with the work concept in mechanics, is that of ideal rolling down an inclined plane. Figure 3 shows this familiar problem. A cylinder or sphere rolls down an inclined plane without any slipping at the interface or point of contact. The interesting point is that although the observer in the stationary frame sees the force P in motion no work is done on (or by) the rolling body. This at first seems to be an anomaly, but the problem may be explained in two ways. First, the motion to and from the point of contact is perpendicular to the inclined plane if there is no distortion. Accordingly, the work is zero even though the force P moves in space; of course, P features in the momentum equation should this alternative method of solution be employed. Another more satisfying explanation of this is afforded using the model shown in Fig. 3(b). At any instant, the force P is acting at the boundary of the system on a particular element of matter within the system. As rolling proceeds, this element is replaced by the adjacent element, so that the rolling may be considered as a series of discrete steps in each of which the work is zero. So, although the force P is observed as moving in space, it does not move attached to material at the boundary of the system. It therefore seems that the definition of work given above requires some refinement with regard to 'the point of application'. (In real rolling, there is some work interaction in the region of the contact, as dry friction is ultimately the result of the motion of many elementary forces acting on elements of the system at the interface. This phenomenon requires a separate treatment and is not pursued any further here.) Attention is now drawn to those examples in applied mechanics which have been chosen to demonstrate how work and energy are evaluated in the most general case when the frame of reference is in motion relative to the Earth.
cylinder or sphere Stationary \frame f '
Fig. 3.
/
//
(a)
Ideal rolling down an inclined plane, (a) general model, (b) model of contact point.
Mechanical energy in different frames of reference
339
EXAMPLES
Linear motion of a particle An obvious starting point is the case of a particle acted on by a constant force. Four frames of reference have been chosen: (a) (b) (c) (d)
the Newtonian or stationary frame; a uniformly accelerating frame; a constant velocity frame; a frame attached to the particle.
Of interest are the velocities and distances moved in these frames (in a specified time), and the kinetic energy change and work done. The results are most conveniently presented in tabular form, and Table 1 lists the various quantities pertaining to unit mass. In all cases, the change in kinetic energy equals the work done in the frame, calculated as the effective force multiplied by the distance moved. The effective force is the real force less the fictitious force where appropriate. The important point to be emphasised is that, in general, work done depends upon the frame of reference. This important conclusion is reinforced by the results for the very special case (d), where the frame moves with the particle itself. Clearly, there is no change of mechanical energy and no work transfer; an observer with the frame sited on the particle would not be aware of any change in the motion of the particle. Although he or she observes a real force, the observer cannot detect any displacement. Finally, it should be pointed out that the data for the simple case shown in Fig. 1 are consistent with the expressions listed in Table l, cases (a) and (b). This may be verified by substitution.
Connected particles A more interesting example is that of two particles connected by an inextensible cord, as shown in Table 2. One 'mass' is in motion on a smooth horizontal table; the cord passes over a frictionless pulley and is connected to the second 'mass' which is in vertical motion. Again, the results are listed in tabular form; only those for the Newtonian frame and constant horizontal velocity frame are shown. It is a straightforward matter to obtain the corresponding results for the most general case of the horizontal accelerating frame. This case, however, requires caution, as, in an accelerating frame, the second mass sustains not only the vertical force of its weight, but also a horizontal fictitious force associated
x frame
force
(½)af2t 2
(½) a2t2
Final kinetic energy of unit mass
Change in kinetic energy of unit mass
am
am
(at - am)[-Umt + (½)(af - am)t 2]
[-um + (af - am)t]2~2
(½)(--Um) 2
(af -- am)[--Umt + (l/2)(af -- am)t 2]
ar-
- u m + (af - am)t -Umt+(l/2)(af - am)t 2
af-
--Um
Uniform acceleration, a m
af[-Umt+ (1~2)aft2]
(-urn + aft)2~2
(1/2)(-- Um)2
af[--Umt + (½)aft 2]
0
0
0
0
0
0
--Um + (l/2)aft2
a~
0 0
0
Frame attached to mass
- u m + aft
a~
--Um
Zero acceleration
Moving Frame (initial velocity, urn)
For the fixed frame and the frame moving with uniform velocity, the acceleration and the force of unit mass are the same according to Newton's principle of relativity (see Ref. 6).
0
(]/2)a2t 2
af"
aft ('/z)af t2
a~
0
Fixed frame
Initial kinetic energy of unit mass
Work done
(Effective) force unit mass
Distance moved
Final velocity of mass
Acceleration of mass
Initial velocity of mass
ftn
unit mass
TABLE 1 Linear M o t i o n of a Particle
L~
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M e c h a n i c a l e n e r g y in different f r a m e s o f reference
TABLE 2 Connected Particles
( Newtonian )
Fixed frame
Moving frame (constant velocity urn)
Initial velocity of m
0
-- Um
Initial velocity of M
0
-- Um
Mg/(M + m)
Mg/(M + m)
ITIaSS m
frame
mass M
Acceleration of m and M Tension force P
Mmg/(M + m)
Mmg/(M + m)
Final velocity of m
Mgt/(M + m) Mgt/(M + m)
- u m + Mgt/(M + m) {(-urn) 2 + [Mgt/(M + m)]2] 1/2
MZg2tZ/[2(M + m)]
P . s m - P . s + Mgs
Final velocity of M Work done Initial kinetic energy of m
0
(V2)m(-um) 2
Initial kinetic energy of M
('A)M~ -urn) 2 (V2)m[-u~ + Mgt/(M + m)] 2
Final kinetic energy of m
0 MZmg2t2/[2(M + m) 2]
Final kinetic energy of M
M392t2/[2(M ~- m) 2]
M{(-Um)-' + [Mgt/(M + m)]2}/2
Change in total kinetic energy
M2g2t2/[2(M + m)]
(-umMmgt)/(M + m) + M2g2t2/(2(M + m)t
Distance moved by m (= Sm)
MgtZ/[2(M + m)]
- u ~ t + Mgt2/(2(M + m))
Vertical distance moved by M (=- s)
MgtZ/[2(M + m)]
Mgt2/[2(M + m)}
with the acceleration of the frame in that direction. (For the moving frame with constant horizontal velocity, there are no force effects associated with the second mass in this direction.) Despite the somewhat lengthy algebra, it may be shown that the work done is also different in magnitude, but again equal to the change of kinetic energy. There are now three components of work done (and three for kinetic energy change), corresponding to the motion of the mass m in the horizontal, and the motions of the mass M in the vertical and the horizontal directions. Of course, both work and kinetic energy are scalar quantities, and the components may be added algebraically to obtain the total energy transfer and energy change. The choice of the direction of motion of the frame of reference is arbitrary, and although the horizontal has been selected for present purposes, the calculation may be effected with, say, a vertically accelerating frame. In that case, two components of work (and energy) would be associated with the mass m, and one with the mass M. This example then is a further demonstration that in the work-energy principle in mechanics, the work and energies are calculated with respect to the frame of reference. Unlike 'real force', work and energy are not absolute, and are frame dependent.
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Ideal rolling down an inclined plane The examples chosen so far refer to linear motion of bodies and the appropriate dynamics pertaining to those situations. The principles involved must be general and, as an illustration of this, a simple example of combined translation and rotation will now be considered. Reference was made above to ideal rolling with regard to the definition of work. This example also serves well as an illustration of the dependence of work on the frame of reference. Following the practice adopted for the previous examples of tabulating the results for various frames of reference, Table 3 has been compiled for the ideal rolling of a cylinder with a stationary frame and a frame moving in translation with the axis of the cylinder. The solution for this problem in the stationary frame does not require any elaboration, as it is a standard mechanics problem frequently used to demonstrate the work-energy principle. The results for the moving frame, however, deserve comment. Both linear velocities and accelerations are zero in this case, leaving rotational kinetic energy as the only component of energy to be considered. The point now is that the rotational dynamics may TABLE 3 Ideal R o l l i n g D o w n an Inclined Plane
sin
mass M, unit radius
Fixed frame (Newtonian)
114ovingframe (in translation attached to the axis of the cylinder)
0 2g(sin 0)/3 2g(sin O)t/3 2g(sin O)t/3 g(sin 0)t2/3
0 g2(sin2 0)t2/3 g2(sin2 0)t2/3
0 0 0 2g(sin O)t/3 0 g(sin 0)t2/3 g(sin 0)/3 0 g2(sin2 0)t2/9 gZ(sin20)t"/9
(linear and rotational)
(rotational)
1rame
Initial velocity Linear acceleration Final linear velocity Final angular velocity Linear distance (in frame) Angular distance Force F of unit mass Initial kinetic energy of unit mass Final kinetic energy of unit mass Change in kinetic energy of unit mass Work done/unit mass (effective force × distance in the frame, or effective torque × angular distance)
g(sin 0)t2/3 g(sin 0)/3
g(sin O) x g(sin O)fl/3 g(sin 0)/3 x g(sin 0)t2/3
Mechanical energy in different frames of reference
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be calculated independently of the linear ones. As there is no rotational acceleration of the moving frame, the 'real' force P is to be used in the torque in the rotational dynamics of the body. Of course, the force P now does work to produce the rotational kinetic energy, as there is motion relative to the frame. Clearly, should the frame both move and rotate with the rolling body, all the pertinent quantities would be zero as for the simple case of a flame attached to the mass in Table 1. The most general case of a linearly and rotationally accelerating frame (comparable with that listed in column 3 of Table 1) is academic in the extreme and has not been pursued here. However, its investigation would involve the principles employed in the connected-particle example referred to above, with, of course, the additional rotational equations of motion allowing for rotational acceleration.
DISCUSSION A N D CONCLUSIONS The principal intention at the outset of the investigation reported here was to ascertain that mechanical work and hence energy change are dependent on the frame of reference in which the measurements are made. Mechanical work is an energy interaction on a system, involving (in general) changes in both thermal and mechanical energies of the system. By considering an insulated system, the complexities of quantifying changes in the thermal-energy component may be avoided and the net mechanical-work transfer then equals the change of total mechanical energy (i.e. the sum of the potential and kinetic energies). This simple 'work-energy principle' may be used to advantage to demonstrate qualitatively the dependence of work on the frame of reference. Clearly, the potential and kinetic energies of a system are themselves frame dependent, as position and velocity are measured in the frame, and so the total mechanical energy is specified uniquely. Now the 'work-energy principle' is considered as a universal physical law as a consequence of relativity considerations. Accordingly, the energy interaction, or so-called work, must also be dependent on the frame of reference. Although the above reasoning may be made without reference to a specific situation or event, it is satisfying to test such reasoning quantitatively. This has been done in the present work by examining a number of simple dynamics problems. The problems chosen conform to the conditions pertaining to the simple 'work-energy principle', i.e. thermal effects are absent. Furthermore, these problems have already been introduced at an early stage in one's academic training, and so will be familiar to all readers. The three examples chosen are the dynamics of (1) a single
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particle, (2) a system of connected particles and (3) ideal rolling of a body down an incline. Various frames of reference (both inertial and non-inertial) have been considered, and example (3) has been selected to introduce rotational aspects of the problem. In all cases, the kinematics have been solved first, and then the total energy change calculated from the point of view of the observer in the particular frame. For the determination of the mechanical work transfer, the product F . s has been evaluated, but, of course, the force F (or forces where appropriate) is that according to whether the frame is inertial or non-inertial. In the latter case, the force is the effective force, or the real force less the fictitious or inertial force, in order that Newton's law shall hold in that frame (see eqn (2)). Of course, the displacement s is that measured in the chosen frame for compatibility with the procedure for evaluating the kinematics of the problem. It has been shown conclusively, that, if the calculations are made in this manner, the 'work-energy principle' is satisfied irrespective of the frame, as it should be. As to the relative magnitudes of the work in each of the cases considered, that may be obtained from the tables listing the results. In fact, in some cases it is possible to have zero work and, of course, zero energy change. Finally, although this investigation has shown the dependence of work transfer on the frame of reference in a particular area of study, the philosophical implications are completely general. This generalisation is reflected in the following quotation, which is to be found in The Mechanical Universe by Frautschi et al.: s 'From the beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer (moving relative to the earth), everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest' (Albert Einstein, Autobiographical Notes (1949)).
REFERENCES 1. Roberts, J. W., Aspects of work in mechanics. Int. J. Mech. Eng. Educ., 16(1) (1986) 1-16. 2. Lewins, J. D., Letter to the Editor. Int. J. Mech. Eng. Educ., 17(2) (1986) 131-5. 3. Bridgman, P. W., The Nature of Thermodynamics. Harper Torchbook, London, 1961. 4. Gibbings, J. C., Thermomechanics. Pergamon Press, Oxford, 1970. 5. Barrow, H. & Pope, C. W., Clarification of the evaluation of friction work transfer in control volume analyses. Int. J. Mech. Eng. Educ., 19(1) (1991).
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6. Symon, K. R., Mechanics. Addison Wesley, Reading, Massachusetts, USA, 1973. 7. Reynolds, W. C., Thermodynamics. McGraw-Hill, New York, 1968. 8. Frautschi, S. C., Olenick, R. P., Apostol, T. M. & Goodstein, D. L., The Mechanical Universe, Mechanics and Heat, Advanced edn. Cambridge University Press, London, 1986.