Classical Mechanics -- Sign of work done

Question

It seems that work has two possible ways to decide it's sign:

Whether you take the perspective of the system or the surrounding (whether you consider work done on the system as positive, or work done on the surroundings as positive)

And also whether the component of force parallel to motion is in the same direction as your motion (if you move in the same direction as motion, it is positive work and if in the opposite direction, negative work).

While the first definition is commonly seen in thermodynamics and chemistry, the second definition seems to be used in classical mechanics and motion problems (like the work done dropping/lifting an object from some height).

Are these two definitions the same thing, and if so how can they coexist?

So to give an example that makes my question clear:

When you say that the work done in dropping an object from 3 meters to 0 meters off the ground is positive, are you taking the perspective of the system or the surroundings in this case? What is the system and what is the surroundings (I think one of them is the gravitational field), and are you considering the work done by the system to be positive or negative? How does the direction of force relative to motion play also affect the sign?

Answer 1

In mechanics, a mass $m$ experiences a force $\textbf{F}$ along some path $C$. The work done on the mass is given by

$$ W = \int_C \textbf{F} \cdot d\textbf{r},$$

such that the energy of the mass increases by $W$. Positive work corresponds to energy being added to the system in question (which is inevitably taken from the surroundings).

Edit: To answer in terms of the example, when a mass falls near Earth's surface, it experiences the locally downward force of gravity. This force is in the same direction as its motion, so we say that positive work is being done on the mass.