Positive work while lifting an object

[sawer]

According to work formula
$$
W = \mathbf F\cdot\Delta\mathbf x
$$
and
$$
W = F\Delta x\cos\theta
$$
If an object falls:
F = Gravitational Force = negative
Delta x = Final Position - Initial Position = negative ==> (Like 2 - 5 = -3 ; because it is falling)
cos 0 = 1 ==>(because x and F have same direction)

And work is = negative.negative.positive = POSITIVE
The work is done by gravitational force for falling body is positive. Right?

But if an object rises
F = Gravitational Force = negative
Delta x = Final Position - Initial Position = positive==> (Like 5 - 2 = 3 ; because it is rising)
cos 180 = -1 ==>(because x points upward and F points downward direction)

And work is = negative.positive.negative = POSITIVE
The work is done by gravitational force for lifting body is still positive.

But it must be negative. Right?
So my question is: What is wrong?

 

[Dale]

In the second formula $F$ and $\Delta x$ are the magnitudes of the corresponding vectors in the first equation. Thus they are never negative. In the second formula the only term that can ever be negative is the cosine term.

 

[sunmaggot]

Thank you. I got it.

But one thing.

Isn't $\Delta x$ ==>> final - initial position. So why can't it be negative?

the definition of positive work done: some work done on an object which increases its energy. for example: acceleration, object gains more energy by work done.
the definition of negative work done: some work done on an object which decreases its energy. for example: deceleration, object loses energy by the work done.
please note that, one object does positive work done on another object, say A make positive work done on B, then B makes negative work done on A, they are symmetric.

This is my understanding, maybe there is mistake, but so far I think this definition is okay.

 

[theodoros.mihos]

As @DaleSpam say:
$$ W = \mathbf{F}\cdot\mathbf{r} = |F||r|\cos{\theta} $$
so take the sign by angle and all other are positive.

 

[nasu]

the definition of positive work done: some work done on an object which increases its energy. for example: acceleration, object gains more energy by work done.
the definition of negative work done: some work done on an object which decreases its energy. for example: deceleration, object loses energy by the work done.

This will be true if you specify that you are talking about kinetic energy and not energy in general and that is the work of the net force .

When an object is raised up, gravity does negative work but the potential energy of the body increases.
The resistance force on an accelerating object does negative work but the KE of the object increases.