Why is potential energy negative?

[Hertz]

Total energy is $\frac{1}{2}mv^{2}-mgh=C$ for a classical non-rotating object right? C can be determined based on an arbitrary initial condition that suits the problem, right? How come it's not $\frac{1}{2}mv^{2}+mgh=C$? This seems like a more suitable definition of "total" energy. I'm sure there's a reason it's minus and not plus, but what is the reason?

[edit]
Does it have something to do with the direction of the conservative force field that the object is in?

 

[mfb]

It just depends on the definition of h. If h is upwards, it is +mgh, otherwise it is -mgh.

 

[jtbell]

Total energy is $\frac{1}{2}mv^{2}-mgh=C$ for a classical non-rotating object right?

According to who? That's not what I see in the textbooks that I've used, which all have a + sign, and define h as increasing in the upwards direction.

 

[chandra.phys]

Total energy is $\frac{1}{2}mv^{2}-mgh=C$ for a classical non-rotating object right?

If you have taken the surface of the Earth as the reference point for your potential energy, then the potential energy of a body of mass m at a height h (small compared to Earth's radius) is mgh. It is a positive quantity with respect to the reference point. The total energy in such a case is $\frac{1}{2}mv^{2}+mgh$. The Lagrangian can be $\frac{1}{2}mv^{2}-mgh=L$.

 

[PJC01]

Yes, the reason potential energy is negative in this equation is due to the direction of the conservative force field that the object is in. When an object is in a gravitational field, the force exerted on it is always in the direction of decreasing potential energy. This means that as the object moves closer to the source of the gravitational field, its potential energy decreases and becomes more negative.

In this equation, the term -mgh represents the potential energy due to gravity. Since the gravitational force is always in the direction of decreasing potential energy, it makes sense for this term to be negative. This negative sign is necessary to ensure that the total energy of the system is conserved, as it takes into account the change in potential energy as the object moves through the gravitational field.

Furthermore, potential energy is defined as the energy an object has due to its position or configuration in a force field. In the case of a gravitational field, it is the energy an object has due to its position in the Earth's gravitational field. Since this energy is relative to a reference point (usually chosen as the ground level), it is important to define it as a negative value in order to accurately represent the energy of the system.

In summary, the negative sign in the potential energy term is necessary to accurately describe the energy of the system and account for the direction of the conservative force field. It is a fundamental aspect of the equation and cannot be changed to a positive sign.