Basic Thermodynamics ~ J. Pohl © (1/19/14) | www.thermospokenhere.com (C256a) (C256) |
A student, having given some thought to a quite simple event of a Block, used definitions of physics along with Newton's 2nd Law and the concept of potential energy to arrived at two, contradictory conclusions. The situation, arguments and conclusions (in the student's words) are presented below.
Note: This discussion is no way meant to be a "correction" of anything. No "answer" will follow. The discussion is about definitions, differences of opinion, and differences of perspective.
The student's written statement is below. Beneath that a discussion: What was said and what does it mean.
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(S)tatement: Suppose a block is being raised at constant velocity by a tension in a cord against the force of gravity. Is there any work done on the block? [underline added]
Argument/Conclusion (1): My first answer was "no," since Work = Force*Distance, Force = Mass*Acceleration, and if velocity is constant then acceleration is zero and force equals zero. So then work is zero.
Argument/Conclusion (2): However work also comprises mgΔH, and if the height is changing then there is an increase in potential energy. So my second answer is yes there is work.
Please resolve this contradiction.
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First, reading only the Statement, can the following aspects/meanings be agreed upon (as beginning points)?
Scenario: The physical scenario, the model of reality, is a Block as it experiences its event.
Comment: This seems clear.
System: The system is the Block (only).
Comment: While there is no specific specification of system, the statement implies
that the Block alone is the system. The Block moves upward at constant velocity as
two
forces act: T, tension of the cord (acting upward) and Fg, gravity force acting downward.
Question: The request is specific and select: "Is there any work done on the Block?"
Comment: Any and only, "work done on" is requested. "Work done on" the Block
happens when a force acting on the Block (directed upward) displaces upward.
If agreed that the above is what the discussion is about, below we discuss perspectives.
The student analyses the system from two vantages (1) and (2)
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Argument/Conclusion (A/C-1) | Discussion |
"...since Work = Force * Distance, and Force = Mass*Acceleration, and if velocity is constant then acceleration is zero and force equals zero. So then work is zero. |
i) Since "force" in Newton's 2nd Law means sum of forces (ΣF) and there being two forces germane (T and Fg), is work = ΣF * Distance correct? Does it follow that the conclusion of this argument is: ΣW = 0. ii) We seek "work done by." How does ΣW = 0 related to "work done by"? |
Summary: Is this correct but the conclusion is broader; ΣW = 0 But the "work" to "work done on" relation is not explained. |
This author is writing this in hope of clarifying the issues. The student takes conclusions from (A/C-1) and compares them to (A/C-2). There is a marked difference in perspective between (A/C-1) and (A/C-2). It is that their systems are not the same. Physical reality seems the same, but the analytic perspective is not. Physical conclusions must be invariant with perspective of study.
Explanation of Perspectives:
(A/C-1) Block as System: The Block is modelled as a point mass (or extended BODY). Forces of tension of the cord and gravity are identified explicitly.
Though Earth is nearby, the student in using "force of gravity," chose Earth to be surroundings. Forces are the mechanisms whereby surroundings influence system events. When discussion of a system includes a force attributable to some "X", that "X" is surroundings.
(A/C-2) Earth and Block as System (2): Gravity work (the force of gravity displaced) is special - it is path independent. Knowing mass, one can see the gravity work as sure as one can see elevation changes. Work is energy change but this being a matter of elevation, the thought was to associate it with the "small part of the system," the mass. And to call it an energy effect not a work effect. The understanding of potential energy commenced some 150 years after Newton's death. Wait a minute. Also needed was experimental determination of the surface acceleration of Earth, goIts more. There's anumber need... surface acceleration of gravity.
Argument ~ Conclusion (A/C-2) | Discussion |
"However, work comprises mgoΔH, and height is changing then there is an increase in potential energy. So my second answer is yes there is work." Please resolve this contradiction. |
i) There is a fundamental different, (2) compared to (1). Mass is the same but with (2) there is more information: the constant, go and the variable measurement, H, as height of the mass above Earth with the implied datum of measurement being the Earth surface. Does Analysis (2) start with the same info as Analysis (1)? If there is a change of perspective, what is it? |
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Author 1: This argument by intersplice of definitions concludes "The work of a Block that moves at constant velocity equals zero" appears sound but how is it relevant to the question asked?
(i) "Force equals mass times acceleration" is true in a dimensional sense. That is [F] iden; [m][L][t]-2. If "work equals zero" is then "work done on equal zero," or what?
This argument is imprecise. Does it address the Block? For the Block: F ≠ mA but ΣF = mA.
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Student 2: "... but work comprises mgoΔH,... if height is changing... conclude "work done on" is NOT zero.
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Author 2: "work" does not "comprises mgoΔH." Work and To indtoduce potential energy (being an Earth-and-Block as system idea) is to return to change the system from Block (only) to be Earth and Block (combined as system). What has change of potential energy to do with "work done on?" Please show effort, show steps of your logic.
Also The interest is "work done on." To head this way, please show what "work done on" has to do with is, and how the "work" as
in the above equations being zero parses to conclude "work done on" is zero.
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Author's Comments: The student is clear in asking for "work done on."
The student's statement is unclear. Work is not germane for this "steady-continuing" event. "Work-rate" is the proper work-related form. Work (and work-rate) as ideas arrived to engineering physics about the time Watt and Newcomen developed the first steam engines. The student does not define the system for either analysis. What system, the reader must deduce.
Conclusion 1: The student is correct. The work-rate of the Block is zero as a sum of two equal and opposite work-rates in action.
Conclusion 2. The student, by mention of potential energy, is using "Block and Earth" as the system. A work-rate occurs, it is the tension of the cord times its vertical velocity. This work-rate equals the rate of increase of potential energy of the Earth/Body system.
Solution 1: For the first analysis, the student uses the Block as system. Also he makes the classic algebra-based-physics mistake. His use of "F" in Newton's 2nd Law causes him to overlook a force. One should use ΣF in the 2nd Law, as Newton intended.
So our task: Block as system, multiply Newton's 2nd Law by Block velocity. Obtain relation for work-rate.
Newton's 2nd Law addresses momentum of the Block in our "0Z" coordinate system. Best ALWAYS to use ΣF. |
![]() | (1) |
Mass of the Block and its vertical velocity are constant so left-of-equality equals zero. |
![]() | (2) |
Expand ΣF to establish gravity force as equal and opposite to the cord force. This is obvious but needed later. |
![]() | (3) |
Return to Eqn-2, multiply the equation by the vertical velocity of the Block. |
![]() | (4) |
Now expand the sum of forces. |
![]() | (5) |
Write the forces as magnitudes and directions. |
![]() | (6) |
The vector multiplications determine the signs: |
![]() | (7) |
Force times velocity equals work-rate. The sum of these work-rates is zero. |
![]() | (8) |
Conclusion 1: The student is correct. The work-rate of the Block is zero as a sum of two equal and opposite work-rates in action.
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Solution 2. The student's statement "work equals mgoH" is not applicable from the aspect work. Work-rate as "work-rate = d(mgoH)/dt" is relevant. The mention of increasing potential energy reveals the system to be the Block and Earth taken together. This development below addresses the Block initially with Earth participating via gravity force. Block and Earth are brought together near the end.
Newton's 2nd Law addresses momentum of the Block in our "0Z" coordinate system. Best ALWAYS to use ΣF. |
![]() | (9) |
In this and a great many instances, gravity force is relevant. Remove the force of gravity from the sum of forces. |
![]() | (10) |
As before, multiply Eqn-9 by the velocity of the Block. Three terms are obtained. |
![]() | (11) |
By basic definitions of calculus, the term left-of-equality Eqn-11 is altered as follows: |
![]() | (12) |
Right-of-equality, the first term changes as follows: |
![]() | (13) |
Since zo is a constant, dzo/dt = 0. The term is "(z - zo)" as a convenience. Place the results, Eqn-12 and Eqn-13 into Eqn-11. Thus with a little grass-roots calculus, Newton's 2nd Law multiplied by velocity, can be represented as Eqn-14.
Important: Although Newton's 2nd Law has been transformed, the system it addressed initially has not changed. That system is the Block. The terms of Eqn-14, left to right are: A modification of momentum, the work-rate of the gravity force and the work-rate of all other forces. This form, known to many persons in the years after Newton's death, displays a new idea, energy of a BODY.
![]() | (14) |
A Body has characteristics (position and velocity) and properties (mass, momentum for now - energy soon). Supposing the mass of a Body is known, for any event its position and velocity can be made quantitative, is observable, and measurable from an external non-intefering reference (called extrinsic, by some). The terms of Eqn-14 are specified simply in terms of velocity and elevation relative to Earth.
![]() | (15) |
When we move the gravity work-rate term from right-of-equality to left-of-equality, the effect is change our system from "Block" to "Block and Earth."
![]() | (16) |
As a last step we define kinetic energy as KE, and potential energy as PE. Substitute these definitions into Eqn-17.
![]() | (17) |
We can group the terms within the differentiation.
![]() | (18) |
The above concludes the development.
Conclusion 2. The student, by mention of potential energy, is using "Block and Earth" as the system. Consequently Eqn-16 applies. Since the vertical speed is constant, the first term is zero - Eqn-19. A work-rate occurs, it is the tension of the cord times its vertical velocity. This work-rate equals the rate of increase of potential energy of the Earth/Body system.
![]() | (19) |