Solving the Work-Energy Theorem Error in Block-Table System

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Discussion Overview

The discussion revolves around the application of the work-energy theorem to a block-table system where a block comes to a stop on a rough table. Participants explore the implications of energy transformations, the role of friction, and the assumptions underlying the theorem's applicability in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents an application of the work-energy theorem, concluding that the net work done on the block-table system is zero, leading to a contradiction.
  • Another participant questions the assertion that the work done by friction is zero, highlighting the inconsistency in the argument regarding the table's motion.
  • Some participants suggest that energy does not disappear but transforms into heat, with one noting that the block may not come to a complete stop.
  • There is a proposal that if the table is anchored, it would absorb some energy, potentially resolving the contradiction.
  • A participant raises the question of how including heat in the equation would affect the overall energy balance.
  • Another participant asserts that the work done by friction is responsible for the energy dissipation, emphasizing the importance of considering heat in the analysis.
  • One participant suggests that the work-energy theorem is not applicable to objects subject to friction, arguing that it only applies to particles.
  • Another participant mentions the conservation of energy, indicating that energy must be converted to another form due to the difference in kinetic energy.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the application of the work-energy theorem, the role of friction, and the nature of energy transformation in the system. The discussion remains unresolved with no clear consensus on the correct approach.

Contextual Notes

There are limitations regarding assumptions about the system's constraints, the treatment of friction, and the definitions of work and energy in this context. The applicability of the work-energy theorem to non-particle systems is also under debate.

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A block (2 kg) is moving with an initial speed of 1 m/s on a horizontal rough table comes to a stop eventually.

Applying work-energy theorem to the block-table system, we obtain

Change in the kinetic energy of the block + Change in the kinetic energy of the table = Net work done on the block-table

(-1 J) + (0) [the table doesn't move] = Net work done on the block + Net work done on the table
= [Work done by normal force on the block by table + Work done by friction on the block by table + Work done by weight on the block] + [Work done by normal force on the table by block + Work done by friction on the table by block+ Work done by weight on the table]

Work done by weight on the block is 0 because the angle between the displacement and the weight is 90 degree. Work done by weight on the table is 0 because it doesn't move and therefore has 0 displacement. Work done by the two frictions is also 0 because friction on block by table and friction on table by block are action-reaction pair and thus in general, their resultant is zero and produce zero work. The same goes to the normal force.

Thus, basically the net work done on the block-table is zero.

And we get:
-1 = 0
What is wrong?
 
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Work done by the two frictions is also 0 because friction on block by table and friction on table by block are action-reaction pair and thus in general, their resultant is zero and produce zero work.

But you just said the table doesn't move.
 
So, the table should move to fix the -1 = 0?
 
The energy doesn't disappear, it becomes heat.
The block and table both start radiating in the infrared.
It is also unlikely the block will come to a complete stop, but will
"bounce" and thereby consume some more energy.
 
Unless something's holding the table in place. In which case the table and that thing get a slight amount of energy which fixes that. If it's anchored to the Earth, assuming the Earth's perfectly rigid (which it isn't, but assuming,) then it'll have its motion changed slightly to fix that.

EDIT: Duh. Heat. *Bangs head against wall due to the sheer stupidity of the fact that I overlooked that*
 
If we include heat in the equation, how will it fix the equation?
Or if the table is free to move, then it will move and gain kinetic energy of 1 J?
 
Work done by weight on the block is 0 because the angle between the displacement and the weight is 90 degree.

No, the work is done by the force of friction which acts opposite to the displacement vector.
If we include heat in the equation, how will it fix the equation?

There is one joule of heat disippated. This is the work done by the force of friction stopping the motion of the block.
 
I have done some reading to find the answer. This is what I got:

1. Work-energy theorem applies only to particles.
2. Objects subject to friction cannot be treated as particles.
3. Thus it is wrong to apply work-energy theorem to objects subject to friction.
 
From conservation of energy, there is difference in initial and final kinetic energy.
So the energy must have been converted to other form, conservative or non-conservative.
 

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