I was reading some notes that offer the standard definition of energy,
which is the ability to do work. Consider an elevator that is going
up. That elevator is doing work because it is exerting force in the
same direction as the displacement. My question is: how does this
work relate to energy? As the elevator goes up, its total potential
energy, $i.e.$ $m\times g\times h$ is increasing. However, is this
energy separate from the energy required to do the work in
the first place? Os is the correct reasoning that as the elevator
is rising up, it is generating exactly enough potential energy to
exert force to go back down? I am not a physics student and I find
energy super difficult to grasp. Any help is much appreciated.
Newtonian Mechanics – Work and Energy in an Elevator
energynewtonian-mechanicswork
Related Solutions
Sometimes when you're stuck on things, it's helpful to look at the mathematics of what's being asserted. For example, nowhere in Newton's three laws does "energy is conserved" appear.
Energy conservation does appear, however, when you have a system that behaves like $m \ddot{x}=-\nabla U$, for some function $U$, where $x$ is a position vector as a function of time. In this case it's a mathematical theorem that $\frac{d}{dt}\left(\frac{1}{2} m \|\dot{x}\|^2+U\right)=0$.
Though it's easy to get carried away and start talking about nature and systems and why some forces can be represented as $\nabla U$, in every regular mechanics book* I've read, this is what things boil down to.
*regular mechanics as opposed to higher mechanics. In higher mechanics one states that the action $A[u]=\int L(u(t),u'(t),t)dt$ tends to be minimized. From that it's a mathematical theorem that if $L(u,\dot{u},t)=L(u,\dot{u},t+t_0)$ for all $t_0$, then energy is conserved. However then your question becomes, "why does nature tend to minimize the action" or equivalently, "why must we use a function like $L$?" To which one must appeal to experiment! There are no proofs of energy conservation just as there are no proofs of Newton's laws!
Don't be surprised that physics has a lot of definitions that are circular. Ultimately, we are just describing the universe.
Work and energy have been defined in a certain way in newtonian physics to explain a kinematic model of reality. This is a model, not reality - you will find no such thing in reality. However, in many scenarios, it is close enough to reality to be useful.
For example, let's say that a human has a 10% efficiency at converting food to mechanical work. So if you spend 1000 kJ of food energy to press against a wall, are you doing 1000 kJ of work, or 100 kJ of work, or 0 kJ of work?
In strict mechanical sense, you did no work whatsoever, and all of the energy you used was wasted as heat. If you instead used this energy to push a locomotive, you would have wasted "only" 900 kJ of the energy as heat, with 100 kJ being work. But the locomotive has its own friction, and it wil stop eventually, wasting all the energy as heat again. And overall, you did expend all those 1000 kJ of food energy that is never coming back.
All of those are simplifications. Kinematics is concerned with things that move. Using models is all about understanding the limits of such models. You're trying to explain thermodynamics using kinematics - this is actually quite possible (e.g. the kinematic theory of heat), but not quite as simple as you make it. Let's look at the fire example. You say there is no displacement, and therefore no work. Now, within the usual context kinematics is used, you are entirely correct - all of that energy is wasted, and you should have used it to drive a piston or something to change it to useful work.
Make a clear note here: what is useful work is entirely a human concept - it's all 100% relevant only within the context of your goals; if you used that "waste" to heat your house, it would have been useful work. It so happens that if you look closer, you'll see that the heat from the fire does produce movement. Individual molecules forming the wood wiggle more and more, some of them breaking free and reforming, and rising with the hot air away from the fire, while also drawing in colder air from the surroundings to feed the fire further. There's a lot of displacement - individual molecules accelerate and slow down, move and bounce around... But make no mistake, the fact that kinematics can satisfactorily explain a huge part of thermodynamics is just a bonus - nobody claimed that kinematics explains 100% of the universe. It was a model to explain how macroscopic objects move in everyday scenarios. It didn't try to explain fire.
For your specific questions, you really shouldn't ask multiple questions in one question. It gets very messy. But to address them quickly:
- There is no potential energy in the kinematic model. The concept is defined for bound states, which do not really exist as a concept in kinematics. In other models, you might see that there's a difference between, say, potential energy and kinetic energy - no such thing really exists in reality. You need to understand the context of the model.
- In a perfectly kinematic world, this is 100% correct. However, as noted before, kinematics isn't a 100% accurate description of reality, and there are other considerations that apply, such as the fact that humans have limited work rate, limited ability to apply force, and the materials we are built of aren't infinitely tough, perfectly inflexible and don't exist in perfect isolation from all the outside (and inside) effects. In real world applications of models, these differences are usually eliminated through understanding the limits of given models, and using various "fixup" constants - and if that isn't good enough, picking (or making) a better model.
- You're mixing up many different models at different levels of abstraction and of different scope so confusion is inevitable. Within the simplified context of kinetics, there is no concept of "potential energy". You simply have energy that can be used to do work, and that's it; it doesn't care about how that energy is used to do work, about the efficiency of doing so etc. In another context, it might be very useful to think of energy and mass as being the same thing - and in yet another, they might be considered interchangeable at a certain ratio, or perhaps in a certain direction, or at a certain rate. It's all about what you're trying to do.
- How is that equation useful? That's the only thing that matters about both definitions and equations. I can define a million things that are completely useless if I wanted to - but what's the point?
- Within the original context, those aren't considered at all. Within a more realistic context, both heat and sound are also kinematic.
The reason you have so much trouble finding the answer to your questions on physics sites and forums is that the question doesn't make much sense in physics. It's more about the philosophy of science, and the idea of building models of the world that try to describe reality to an approximation that happens to be useful to us. You think that those words have an inherent meaning that is applicable in any possible context - this simply isn't true. From the very inception of the idea of physics, people have known that it isn't (and never will be) an accurate representation of reality; and we've known for a very long time that, for example, different observers may disagree on the energy of one object. You just need to understand where a given model is useful, and pick the right model for the job. Don't try to drive a screw with a garden rake.
Best Answer
Since you are new to physics, in order to help answer your questions the following is a brief primer on work and energy.
The relationship between work and energy is that work is one of the mechanisms by which energy is transferred from one thing to another. In the case of work, the transfer of energy is due to force times displacement. (The other main mechanism is heat. Heat is energy transfer due solely to temperature difference).
If the direction of the force is the same as the direction of the displacement, the work done by the force is called positive work because it transfers energy to the object being displaced. This is the case with the work done on the elevator. A motor does work on the elevator (and its contents) by lifting it up with a cable against the force of gravity. The motor transfers energy to the elevator and its contents.
If the direction of the force is opposite to the direction of the displacement, the work done by the force is called negative work. When negative work is done by the force it takes energy away from the object being displaced. In this case when the elevator rises gravity does negative work on it since its force is opposite to the displacement of the elevator. Gravity takes energy away from the elevator and its contents and stores it as gravitational potential energy of the earth-elevator system. More on this below. Gravity is what is called a conservative force. That means the work done by gravity only depends on the initial and final positions of the object, independent of the path between the positions.
It is important to understand that it is not the potential energy of the elevator that is increasing. It's the potential energy of the combination of the elevator and the earth (earth-elevator system) that is increasing. That's because potential energy is a system property. Although you will commonly see people refer to the potential energy of objects, that is technically incorrect. The object without the earth and the earth without the object experiences no change in gravitational potential energy due to the movement of either.
That said, the increase in potential energy is not seperate from the energy required to do the work in the first place. Thats because the origin of the energy that gravity took away from the elevator to store as gravitational potential energy is the energy transferred to the elevator by the motor. But that does not necessarily account for all the energy transferred to the elevator by the motor. Part of that energy may be the change in kinetic energy of the elevator. To explain:
A. As the elevator passes a floor a height $h$ on the way up gravity takes energy of $mgh$ away from the elevator to store as gravitational potential energy. But since the elevator is still moving up, it has kinetic energy of $\frac{1}{2}mv^2$. That kinetic energy equals the difference between the positive work done by the motor on the elevator, minus the energy taken away by gravity (gravitational potential energy).
B. Just before the elevator reaches its intended floor at say height $h'$, it decelerates to come to a stop. At that point its kinetic energy is zero just like it was at the start of the trip. The total change in KE is thus zero. This means that the net work done on the elevator, $+mgh'$ by the motor and $-mgh'$ by gravity, equals zero.
This last point illustrates the important work-energy theorem, which states that the net work done on an object equals its change in kinetic energy.
Hope this helps.