Gravity and mass relation in Conservation of Energy equations

Question

When writing Conservation of Energy equations, if you want to add a gravity component, you write, $mgh$. Where $m$ is mass, $g$ is gravitational constant, and $h$ is the height. If a mass has nothing to do with gravity on falling objects, why is it here?

Young Kindaichi · Accepted Answer

If a mass has nothing to do with gravity on falling objects, why it here?

Its true mass of body has nothing to do with gravitational acceleartion but it has to do with the gravitational force.
We defined the gravitational energy to be the work done by a particle to travel from infinite to that point. For uniform force, We can write the work done by a body when particle travel from point $a$ to $b$ :
$$W_{ba}=mathbf{F}_0cdot(mathbf{r}_b-mathbf{r}_a)$$
for gravity $mathbf{F_0}=-mghat{z}$, so if the particle moves from $mathbf{r}_a$ to $mathbf{r}_b$, the change in potential energy is
$$U_b-U_a=-int_{z_a}^{z_b}(-mg)dz=mg(z_b-z_a)$$

So basically the work done depend on the mass and so is the potential energy. It's obvious you need to do more work if you are uplifting a car than to cycle.

Not_Einstein · Answer

If a mass has nothing to do with gravity on falling objects, why it here?
Not true. Mass has nothing to do with the speed at which an object falls - a small mass, m, and a larger mass, M, when released at the same time from the same height, will hit the ground at the same time (ignoring air resistance). But their kinetic energies when reaching the ground will not be the same: $frac{mv^2}{2}$ < $frac{Mv^2}{2}$ where v is the common final speed.

Gravity and mass relation in Conservation of Energy equations

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