Over one complete cycle, 640 J of heat is put into a heat engine that operates with 50% efficiency. How much work is done by the engine in that cycle?
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In this problem, you are asked about work done by a heat engine. In a heat engine, thermal energy is put in and work is put out. In other words, heat engines can be understood by tracking energy. This is a Conservation of Energy problem.
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Unlike mechanical energy problems, no figure is needed in this case. The heat engine goes through a complete cycle—it returns to its original state. So drawing a picture of the engine will not help you track energy changes.
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Any time you understand the motion of an object by looking at its energy, you begin with a statement of Conservation of Energy.
The First Law of Thermodynamics states Conservation of Energy in the most useful form for heat engines:
ΔQ = ΔU + W
This is the only equation you will need for this problem. -
ΔQ = ΔU + W (1)
320 J = 0 + W (2)
W = 320 J (3)
The heat engine does 320 J of work in one complete cycle. No further mathematical solution is needed for this problem.
How did you get 320 J?
Efficiency is defined as the output divided by the energy put in:
Eff = 0.50 = (Qused to do work)/(Qtotal input) = Q/640 J
0.50 = Q/640 J
Q = 320 J
How did you know ΔU is zero?
Over a complete cycle, an engine returns to its original state. In other words, it has the same properties as at the start of the cycle—the internal energy is the same. Therefore, change in internal energy is zero.
Mechanical energy problems are those that involve a change in velocity as an object changes its position.
ΔQ is the change in the internal energy of the system.
W is the work done by the engine. A positive value of W means that work is done by the engine; a negative value means that work is done on the engine.
Efficiency is defined as the output divided by the energy put in:
Eff = 0.50 = (Qused to do work)/(Qtotal input) = Q/640 J
0.50 = Q/640 J
Q = 320 J
Over a complete cycle, an engine returns to its original state. In other words, it has the same properties as at the start of the cycle—the internal energy is the same. Therefore, change in internal energy is zero.
My text book has separate chapters for thermodynamics and mechanical energy. Why did you call this an energy problem?
The same core physics (energy is never created or destroyed but can change in form) is behind all energy problems, regardless of how the energy changes. Physicists studying motion tend to look at problems in which the key conversions are between kinetic, potential and thermal energies; thermal physicists tend to look at situations in which the key conversions are within thermal energy itself or in which thermal energy is used to do work. In all cases, you begin with a statement of conservation of energy, plug in the energy changes for your specific situation, and proceed in the same way.
Can this problem be worked using kinematics or dynamics?
No. This problem requires you to track thermal energy.
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The energy chain for this problem is
Internal energy is not included on the energy chain because it is the same at the end of the cycle as it was at the start.
We know that 640 J of energy are put into the engine and that half of that value is exhaust (50% efficiency.) Therefore, the remaining 320 J of energy go to do work.