Physics

# The Physics Philes, lesson 15: Release Your Potential

In which gravity works, rubber bands are snapped, and energy is conserved.

As I hope I’ve proven over the past couple of weeks, kinetic energy is fun and interesting. It helps explain how much energy is needed to move an object. However, we only have part of the story. We can’t fully understand the concept of work without understanding potential energy.

Kinetic energy is energy associated with motion, while potential energy is energy associated with a position. I’ll focus today on two forms of potential energy: gravitational and elastic.

Gravitational potential energy is the potential energy associated with a body’s weight and its height above the ground. We can find the work done on an object by gravitational potential energy by using the general work equation: W=Fs. We can calculate the gravitational potential energy by multiplying the weight (which, remember, is a force) by the displacement of the object on the y-axis. The resulting equation is: This equation works even if the object moves up. But in that case, the work would be negative.

The potential gravitational energy can be expressed by multiplying the mass of the object, gravity, and the height of the object: To find the difference in gravitational potential energy, we subtract the final initial gravitational potential energy from the initial gravitational potential energy, the result being ΔU sub grav. This can also be expressed in terms of y displacement. When y increases, the gravitational potential energy increases. When y decreases, the gravitational potential energy decreases. It’s also worth noting that gravitational potential energy is a shared property between the body and the earth (assuming we’re working with stuff on our little blue planet).

The work-energy theorem says that the total work done on a body is equal to the change in the body’s kinetic energy. We can use the same concept to prove that, when the gravitational force is the only force that acts, mechanical energy is also conserved.

If gravity is the only force that acts, then the total work is equal to the initial gravitational potential energy minus the final gravitational potential energy, which is also equal to the total work done by gravity. This can be rewritten as: The sum of kinetic and potential energy (E) is the total mechanical energy of the system. Above you see that the sum of the initial kinetic energy and the initial gravitational potential energy is equal to the sum of the final kinetic energy and the final gravitational potential energy. Each side of that equation can also be set equal to E as the sum of kinetic and potential energy. This means that E is constant and is an example of the conservation of mechanical energy.

That’s all well and good, but what happens when there are forces other than gravity at work, as well? The work done by all forces other than the gravitational force equals the change in total mechanical energy of the system. Or, in mathematical terms: This equation can be written in terms of gravitational potential energy: But what if we need to calculate the potential gravitational energy along a curved path? Short answer: it doesn’t matter. The work done by gravity is only dependent on the displacement over the y-axis. Since we only need to know how far the body moves up or down, we can use the same expression for gravitational potential energy whether the body’s path is curved or straight.

That is gravitational potential energy! Next week we’ll do some problems involving gravitational potential energy, but for right now I’ll move on to elastic potential energy.

Elastic potential energy is the process of storing energy in a deformable body. A body is elastic if it returns to its original shape and size after being deformed. Like a rubber band. It will spring back to its original form after being stretched out. Unlike gravitational potential energy, elastic potential energy is stored in a deformable body, which means it is not a shared property. It’s easiest to explain this type of potential energy by using the massless, ideal spring. The work done by the spring to move one end from an initial elongation to a final elongation is expressed using the following equation: where k is equal to a force constant of the spring.

To calculate the work done by the spring, we need to find the displacement from the initial position to the final position: Elastic potential energy can be expressed by a given quantity at the beginning and end of the displacement. That value is: The work done by elastic potential energy can be found in terms of the change in elastic potential energy, which leads to this equation:

The farther the spring is stretched, the more the elastic potential energy. When the spring relaxes, the elastic potential energy decreases.

If the only force doing work on an object is elastic, we go through a similar mathematical process as we did when the gravitational force is the only force doing work on an object. In this situation, the total work is equal to the work on by the elastic force. The work-energy theorem tells us that the total work is equal to the difference in kinetic energy. This means that, if only the elastic force does work, then: In this situation, as with gravitational potential energy, the total mechanical energy is conserved.

There are situations where both gravitational potential energy and elastic potential energy act on an object. In these situations, the total work is the sum of the work by gravitational potential energy, work by elastic potential energy, and work by other forces. The work by the aforementioned energies are equal to the difference between the final and initial kinetic energy. The work done by all forces other than the gravitational and elastic forces is equal to the change in the total mechanical energy of the system: where U is the sum of the gravitational and elastic forces.

The forces I’ve tried to explain today are known as conservative forces. When energy is conserved, the total amount of mechanical energy is constant. There are four properties of conservative forces:

1. Conservative forces can be expressed as the difference between the initial and final values of a potential energy function.
2. It is reversible, meaning that any energy deposit we bank can be later withdrawn. Like a ball tossed in the air. The energy is converted from kinetic to potential, and back to kinetic.
3. It is independent of the path of the body and depends only on the starting and ending points.
4. When the starting and ending points are the same, the total work is zero.

As you might guess, there are also non-conservative forces. These forces are, um, not conserved. There is no potential energy function. Kinetic friction and fluid resistance are examples of non-conservative forces. When these forces act on an object, there is no way to get back the lost mechanical energy. Non-conservative forces can cause a loss of mechanical energy or an increase. When it cases a loss, it’s called dissipative force. Kinetic friction and fluid resistance are examples of dissipative forces. The chemical reaction in a firecracker that makes it go BOOM is considered a non-conservative force because it isn’t reversible.

Lots of information this week! Next week we’ll do some example problems. I think I have a pretty good handle on gravitational potential energy, but I’m less sure about elastic potential energy. So if I have anything wrong – especially regarding elastic potential energy – let me know!

Until next time…FOR SCIENCE!

Featured image credit: dullhunk

1. BTW, that equation for gravitational potential energy uses the assumption that the Earth is an infinite flat plane. In an infinite plat plane, gravity always points downwards and it doesn’t change with height. Now, that is a great assumption when very close to the Earth’s surface, since you know, everything looks really flat when you are that close to the planet.

But since in reality the Earth is round, gravity decreases as you get farther away. So after a few hundreds of kilometers, g=9.7. Which means there has to be a better gravitational potential energy equation. But for most things in our experience, g=9.8 is really convenient, so we can ignore all the inconvenient truths I demonstrated.

2. You might be able to understand gravitational potential energy and elastic potential energy a bit better with a little calculus. (And yes, it really is just a little calculus.)

First off, I’m going to use the word “field”. The word “field” here just means any quantity which is a function of space coordinates only. So mgy or -1/2 kx&sup2; are functions of space coordinates alone, but mv and mgt&sup2; are not, since velocity and time are not space coordinates.

(In relativistic mechanics, of course, there’s no difference between space and time coordinates. This is Newtonian mechanics, so we will make the distinction.)

So a conservative force is one that is described by a potential energy field.

If this is the case, then the force is just the negative of the derivative of the potential energy with respect to space. So, for example, if:

U(x) = -1/2 kx&sup2;

then:

-U'(x) = kx

Similarly, if:

V(y) = mgy

Then:

-V'(y) = -mg

In each case, if q is a space coordinate, and U(q) is the potential energy, then -U'(q) is the force.

If there’s more than one dimension, then you need to use vectors, which makes sense when you think about it. And that in turn means vector calculus. This can be subtle, but since we’re using Cartesian coordinates, we’ll ignore the subtleties.

Suppose that the potential energy function is:

U(x,y) = -1/2 mkx&sup2; + mgy

That is, there’s a spring-like force in the x direction and a gravity-like force in the y direction. We can calculate the force using partial derivatives.

The partial derivative of U with respect to x is basically the usual derivative, pretending that y is a constant. So:

-∂U/∂x = mkx
-∂U/∂y = -mg

The force in this case is a vector, with mkx being the x component and -mg being the y component.

Yes, the idea can be extended to model non-conservative forces. It’s way beyond the scope of this series.

• OK, that’s weird. The partial derivative symbol &part; worked, but the raise-to-the-power-of two symbol &sup2; didn’t.

3. Oh, I just noticed a mistake. You squared y1 in your K1+U1=K2+U2 equation.

• Whoops! Thanks! I’ll fix that ASAP.