I'm reading "The Mechanical Universe" (Frautschi et al.) to brush up my understanding of physics. So far, I can solve all the exercises and I have no problems with the mathematical parts. But, being about half way through the book, I have the impression that I don't really "get" some basic concepts.
Specifically, in chapter 10 where they introduce energy, they talk about Galileo's experiment with inclined planes and a rolling ball:
Now that we have a definition of work, let's go back and follow energy conservation through the various stages of Galileo's experiment. First, Galileo lifts the ball a height $h$, performing work on it by applying a force to balance the preexisting force of gravity. We say that work is done by the force that Galileo applies, or alternatively that work is done against the force of gravity. In any case overall energy is conserved; the work represents a transfer from the world outside the ball (namely, from Galileo) to the ball.
Since there is no net force, the ball is displaced without accelerating; thus all the work goes into the potential rather than kinetic energy: $W = U = mgh.$ Next Galileo releases the ball, and acting now solely under the preexisting force of gravity, it rolls down the incline. During this stage gravity does work on the ball. Again overall energy is conserved, the work representing a transfer from potential to kinetic energy. Finally, when the ball rolls back up an incline to the original height, energy is still conserved. Here work is done against gravity; it represents a transfer back from kinetic to potential energy.
Some pages later, they write:
But in physics the word work is used more precisely, to describe an energy transfer from one thing to another carried out by a force acting over distance.
A "thing", OK… And then:
If we do no work that transfers energy into our system, the total energy of the system, potential plus kinetic, is conserved.
I guess my confusion here is what "the system" is. In the first quote they talk about a transfer "from the world outside the ball to the ball", so the "system" here is the ball, isn't it? But if Galileo belongs to the would outside the ball (system) and thus transfers energy to it, why isn't the Earth considered to belong to the world outside the ball? Shouldn't gravity transfer energy to the ball or away from the ball? Instead, in this case they say that gravity does work on the ball, but energy is conserved. Is the energy within the ball conserved? Or are we talking about the "ball-Earth system"? Who decides when energy is transfered to a "thing" and when it is only transfered within a system?
Best Answer
You have to be aware that "energy" is just an abstract concept that helps us understand and solve some problems in an easier way. Do not think of energy in terms of effort we (humans) do to perform some "work". These are related, but thinking in that terms will probably lead to dead ends.
The system is whatever you define it to be.
The "work in physics" is best understood via the work-energy theorem $\Delta K = W$. You can read this as "net work done on an object equals change in kinetic energy". The definition of "system" is important in the context of internal and external forces, i.e. the forces that act within the system (internal) and the forces that are exerted by the outside world (external). Note that both internal and external forces can change system kinetic energy. If this is counterintuitive, just think of explosions: before explosion bombs are initially at rest with zero kinetic energy; after explosion there are many fragments with finite (positive) kinetic energy. Hence, internal forces from within the bomb produced (positive) change in kinetic energy $\Delta K$, i.e. internal forces did positive work. On the other hand, only external forces can change system momentum.
Correct, you can say the system is the ball.
Gravitational force does work on the ball. As the ball moves away from the Earth the gravitational force does negative work and vice-versa. This mechanism is abstracted via concept of gravitational potential energy. Remember the extension of the work-energy theorem
$$\Delta K + \Delta U_g = W_\text{other}$$
where $W_\text{other}$ is work done by forces other than the gravitational force. As objects move away from each other, the gravitational potential energy increases $\Delta U_g > 0$, and if no external work is being done $W_\text{other} = 0$ then kinetic energy of both objects decreases $\Delta K < 0$.
Gravitational potential energy is by definition
$$U_g = -G \frac{m_1 m_2}{r}$$
The same equation applies to both objects involved. This means that you need to specify both objects, and in the context of gravitational potential energy they form a system. This does not mean that you cannot treat the ball as a system separate to Earth, it only means that gravitational potential energy is a shared quantity between two objects.
Energy is always conserved. This is one of the fundamental laws, and no experiment conducted so far has proved it wrong
$$\Delta K + \Delta U + \Delta U_\text{int} = 0$$
In a given process, the kinetic energy, potential energy, and internal energy of a system may all change. But sum of these changes is always zero. You can treat the ball as a separate system, the same principle applies.
As I have already mentioned, internal forces can change object's kinetic energy. Remember explosions in the context of conservation of energy. If you say the bomb is a system, then after the explosion the bomb fragments still make the same system. Internal energy from within the bomb is converted to kinetic energy of fragments, but conservation of energy still holds
$$\Delta K + \Delta U_\text{int} = 0$$
In this example, energy has been transferred from within the system from internal energy to kinetic energy.