# Physics C: Mechanics - Negative Energy and Potential Energy Curves

• mush64
In summary, the conversation discusses teaching potential energy curves and the concept of negative potential energy. The speaker is struggling to understand how energy, a scalar quantity, can have a negative value. An expert explains that scalars can have signs and gives an example of how potential energy can be negative in certain situations. They also clarify that potential energy does not have a direction, but the reference point can be chosen arbitrarily.
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!

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.

CPW
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.

CPW
mush64 said:
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".

CPW and vanhees71
jbriggs444 said:
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.

CPW, vanhees71, kuruman and 1 other person
anorlunda said:
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.

CPW and anorlunda

## 1. What is negative energy in physics?

Negative energy in physics refers to the potential energy of a system that is less than its potential energy at its reference point. It can also refer to the kinetic energy of a system that is moving in the opposite direction of its reference point. In both cases, negative energy is a relative measure and does not mean that the energy is "negative" in the traditional sense.

## 2. How is negative energy related to potential energy curves?

Negative energy is represented by the lower portion of a potential energy curve, where the energy is less than the reference point. This can occur when a system is at a lower height or position than its reference point, or when the system is moving in the opposite direction of its reference point. The potential energy curve visually shows the relationship between potential energy and position for a given system.

## 3. Can negative energy be converted into positive energy?

Yes, negative energy can be converted into positive energy. This can occur when a system gains energy and moves from a lower energy state to a higher energy state. For example, when an object falls from a higher height to a lower height, its negative potential energy is converted into positive kinetic energy.

## 4. What is the difference between negative potential energy and negative kinetic energy?

Negative potential energy refers to the potential energy of a system that is less than its potential energy at its reference point. Negative kinetic energy, on the other hand, refers to the kinetic energy of a system that is moving in the opposite direction of its reference point. In both cases, the energy is negative relative to the reference point, but they represent different forms of energy.

## 5. How is negative energy used in real-world applications?

Negative energy is used in various real-world applications, such as in the design of roller coasters and other amusement park rides. These rides use the conversion of negative potential energy into positive kinetic energy to create thrilling drops and loops. Negative energy is also used in the study of quantum mechanics, where it plays a role in understanding the behavior of subatomic particles.

• Mechanics
Replies
14
Views
5K
• Mechanics
Replies
10
Views
1K
• Mechanics
Replies
10
Views
982
• Mechanics
Replies
9
Views
2K
• Mechanics
Replies
4
Views
1K
• Mechanics
Replies
3
Views
2K
• Mechanics
Replies
6
Views
682
• Mechanics
Replies
4
Views
13K
• Mechanics
Replies
4
Views
1K
• Mechanics
Replies
4
Views
11K