I was wondering if I could get some help closing some fundamental gaps in my intuition of work, as it relates to force and distance travelled.
Say we pull a 1kg box along the ground. We pull with a force of 1N parallel to the ground, over a distance of 1 meter. We have a formula which gives us work done as $$W = F\cdot d \cdot \cos\alpha = 1N \cdot 1m \cdot 1 = 1J$$
Now, say we pull the box along the ground with the same force, over the same distance, but we pull at an angle of $\alpha = 60^\circ$. Now, the work we've done is $$W = 1N \cdot 1m \cdot \cos60^\circ = 0.5J$$
So, this is where my intuition breaks down a bit, and I think it is because of a warped understanding of work. I would have expected that while only 0.5J was spent moving the box horizontally, the remaining 0.5J dissipated somewhere vertically. Was it spent acting against gravity, and gravity won?
As a surface level understanding, I would say that the work done can either be a product of force and distance, but it can also be seen as the change in energy states of the box.
For instance, if I lifted the box 1m straight up by applying an arbitrary force straight up, I would have caused its potential energy to rise by 9.81J. But I have also moved it 1m upwards, so I have also done some more (?) work that would be given by the product of force and distance again. It is my understanding that we don't "add" these two calculations up to find total work done, but rather they are two different calculations that yield the same result?
But the work I do using the $W = Fd$ formula (assuming the force is parallel to the travel vector) can vary wildly depending on how much force I apply to the box, right? And if so, how can it always coincide with the change in the box' potential energy?
When we move the box horizontally, we don't change its potential energy. I'm presuming (and this is what I hope to clarify) that if we measured the kinetic energy of the box over time, it would somehow yield the 1J again, even though it starts and ends at zero. Do we just consider the highest kinetic energy change, and that yields 1J?
In the case of vertical movement, we change the box' potential energy which is a measure of how much work we applied to it. But since we also moved the box over a distance, we have two different formulae, and I'm having a hard time consolidating them. If we consider $\Delta E_p = mgh_2 - mgh_1 = 9.81J$ always, and $W = Fd$ which varies depending on the magnitude of the force, how can they always be the same?
I see how since $F=mg$, the latter equation "becomes" the former, so I'm not blind to the mathematical obviousness. As I mentioned early on, my question is on the intuition.