Gravity conservation / gravitational forces

[Jadaav]

Why are gravitational forces said to be conservative ?

What does conservative mean here ?

 

[D H]

Here's the mathy definition: A conservative force is one that can be expressed as the gradient of a potential function. If that's over your head, a conservative force is a force that conserves (does not change) total mechanical energy (kinetic energy plus potential energy).

It might help to look at forces that aren't conservative. Friction is the canonical example of a non-conservative force. Instead of converting kinetic energy into potential energy (or vice versa), friction converts kinetic energy into heat.

 

[DeIdeal]

Conservative forces are forces for which the work done moving a particle under the effect of the said force ($\int \textbf{F}\cdot \textrm{d}\textbf{r}$) is path-independent.

Example: Say a (reasonably point-like) object is lifted from a desk. When looking at the work done, it doesn't matter how much it's moved horizontally, as gravity is a conservative force: The work done is solely determined by the height of the surface and the height of the point the object ends up at. This is not true for all forces. Moving the same object along a surface with friction, like the surface of the said desk, requires more work the longer the path is.

(A vector field $\textbf{F}$ is in fact conservative if any of the following, equivalent conditions are fulfilled: $\nabla \times \textbf{F}=0$, $\exists V$ s.t. $\textbf{F}=-\nabla V$ (the existence of a potential), $\oint \mathbf{F}\cdot \textrm{d}\textbf{r}=0$)

 

[WannabeNewton]

Why are gravitational forces said to be conservative ?

Say you have a ball and you want to move it from height $A$ to height $B$. Just for fun, you take a crazy zig-zag path to get from $A$ to $B$ (pretend you just had tons of coffee and are insanely hyperactive xD). Will the work done by gravity during this process be the same or different than the work done by gravity if you simply moved the ball in a straight line down from $A$ to $B$?

 

[Jadaav]

Thanks a lot D H, DeIdeal and WannabeNewton :)

It's clear to me now.