Physics C: Mechanics - Negative Energy and Potential Energy Curves

[mush64]

I'm currently taking a course where we are working to teach older physics concepts and combine them with calculus.

I was assigned to work on teaching a unit about energy; for the most part, it stays relatively consistent and can be solved algebraically.
Another topic in this unit is Potential Energy Curves, which I understand for the most part: you can derive the force of an object by taking the negative derivative of a potential energy equation, and from there you can identify points of equilibrium while also using conservation of energy to solve for the speed of a particle.

However, while devising a few examples to explain this, the ones that I found all end up having a negative potential energy, and once I went through the class, I wasn't able to explain why this really occurred, because I myself couldn't figure it out either. I've read a few older forums trying to explain this, but it's just not clicking with me.

I understand that reference points can be relative in regards to energy, but in these examples, the total energy is also negative. How can there be a negative potential energy in the first place, and how would it even be possible to have negative total energy, especially is energy is technically defined as a scalar quantity with no direction.

I would really appreciate it if somebody could go into this concept a little bit more in depth.

Thank you!

 

[anorlunda]

If you have a bookcase with shelves, then moving a book to a different shelf changes its gravitational potential energy. You can choose any shelf to be zero P.E. Relative to that, the values can be plus or minus, but the direction of gravitational potential energy (higher shelf higher energy) is the same for all shelves.

 

[kuruman]

Bound systems have negative total energy in the sense that you need to add energy in order to separate the components. To be more specific. Say you shoot a projectile radially out from the Earth's surface. Right after the launch its total mechanical energy is $$E_{tot}=K+U=\frac{1}{2}mv_0^2-\frac{GM_em}{R_e}.$$ The second term is the potential energy referenced so that it's zero at infinity but negative at any finite distance $r$ from the center of the Earth. As the projectile moves farther out the kinetic energy decreases and the potential energy increases (it becomes less negative.) You can see that if the initial sum of kinetic and potential is negative, it will remain negative because the total energy is conserved. This means that the kinetic energy will drop to zero at some finite distance from the Earth. That's known as the "turning point" at which the projectile will fall back to the Earth. The escape velocity is the value of $v_0$ that will make the total energy equal to zero. Ths means that a projectile shot at escape velocity will reach infinity with zero kinetic energy. A projectile shot at greater than escape velocity will reach infinity with some residual kinetic energy.

Scalars are not necessarily greater than or equal to zero. Work done by friction is a scalar and is negative.

 

[jbriggs444]

Summary:: Assigned to teach/explain potential energy curves to class and was unable to answer the question "if energy is scalar, how can it be negative?"

energy is technically defined as a scalar quantity with no direction.

Scalars can have signs. There is no rule saying that they must be positive.

Note well, "scalar" is not a synonym for "magnitude of a vector".

 

[anorlunda]

Scalars can have signs. There is no rule saying that they must be positive.

Note well, "scalar" is not a synonym for "magnitude of a vector".

That's true, but I wager the students are thinking about "Speed is a scalar but velocity is a vector." In that familiar example, speed really is a magnitude.

 

[vanhees71]

If you have a bookcase with shelves, then moving a book to a different shelf changes its gravitational potential energy. You can choose any shelf to be zero P.E. Relative to that, the values can be plus or minus, but the direction of gravitational potential energy (higher shelf higher energy) is the same for all shelves.

i'd not say potential energy has "a direction". It's a scalar and thus has no direction as a vector. Your example is very nice though: You can indeed just arbitrary choose one shelf such that the book has 0 potential energy there. That's because the absolute value of the potential doesn't matter but only energy differences are observable. No putting a book up from this zero-energy shelf (i.e., moving it against the gravitational force of the Earth) means that you need to put in work, and that's now the potential energy of the book, $U=m g h$, where $m$ is the mass of the book, $g$ the (constant) gravitational field of the Earth, and $h$ the height of the shelf measured from the zero-energy shelf. If you put a book down you gain energy from the gravitational field, i.e., the book provides net work and thus as a lower energy than when on the zero-energy shelf, i.e., now its potential energy is negative, $U=-m g h'$. Maybe it's easier to take $z$ as a coordinate with the $z$-axis pointing in the opposite direction of $\vec{g}$ (i.e., pointing "up"). Then $\vec{g}=-g \vec{e}_z$ and $U=m g z$.

If the students know already about gradients you can also say that by definition the potential of a force (if it exists!) is defined by
$$\vec{F}=-\vec{\nabla} U$$
and that indeed in this case
$$\vec{F}=-\vec{\nabla} (m g z)=-m g \vec{e}_z=m \vec{g},$$
as it should be. You can also add an arbitrary constant to $U$ without changing the force, and that's why you can choose the "zero-energy shelf" completely arbitrary without changing the physics content of the potential.