Positive work while lifting an object

1. May 22, 2015

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?

2. May 22, 2015

Staff: Mentor

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.

3. May 22, 2015

sunmaggot

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.

4. May 22, 2015

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.

5. May 22, 2015

nasu

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.