As sweat drips off his forehead, the crowd hushes, seemingly by an unknown force. Even the sweat from his head refuses to make a sound as it drops to the ground: he needs laser focus. With his eyes stuck on the ball, he raises his club back over his head and cracks his hips, bringing the club flying downward. The crowd hears a loud smack, and cheers erupt across the stadium like a dormant volcano awakening after years of silence. The ball flies high, then rushes down, plopping perfectly on the green and sticking a meter from the hole.
Have you ever wondered how your favorite sports work according to the laws of physics? Physics is in everything. It is literally in how we walk, work, move, and even breathe. This article will allow you to be able to explain what forces and rules apply to a golf ball rising in the air and falling to the ground. We will explore a natural phenomenon known as potential energy and how it relates to conservative forces. So, let's tee off.
Fig. 1 - Physics is all around us, even golf can teach us lessons about potential energy!
Potential Energy Meaning
Before we dive deep into the specifics of potential energy, let's start by talking about energy in general.
Energy is the capacity to do work.
Recall that work is equal to force times the displacement. Written as an integral, it looks like this:
$$W=\int_{x_\mathrm{i}}^{x_\mathrm{f}} F(x) dx\mathrm{.}$$
Here, we see that the work can be calculated by doing the force \(F(x)\) times the displacement in increments of \(\mathrm{d} x\) over the total distance we need to cover (from \(x_\mathrm{i}\) to \(x_\mathrm{f}\)). This means that an object, or a system, has more energy, the more it can exert a force over a certain displacement.
The Work-Energy theorem states that the force exerted over a system's displacement is equal to its change in kinetic energy.
We can describe this theorem mathematically as:
$$W=\Delta KE$$
where \(W\) is the work done and \(\Delta KE\) is the change in the kinetic energy. Remember that the kinetic energy is equal to
$$KE = \frac{1}{2}\\mv^2\mathrm{,}$$
therefore, we can rewrite our work equation by substituting the above equation for our kinetic energy. Our new equation would be
$$W=\frac{1}{2}\\mv_\mathrm{f}^2-\frac{1}{2}mv_\mathrm{i}^2\mathrm{.}$$
This is important because it will help us to understand the relationship between potential energy, kinetic energy, and work.
Potential Energy Definition
For us to really move forward in the Potential Energy and Conservative Forces article, we should probably define potential energy.
Potential energy is energy inherent in an object due to its position relative to others or its intrinsic characteristics.
For example, a pencil sitting on the ground would have no potential energy. However, if I were to pick that pencil up and raise it a meter off the ground, it would have potential energy. But why? What changed?
Now that the pencil is suspended in the air, the force of gravity wants to act on it, and the second I let go, that pencil will come crashing down. Therefore, when I raise the pencil above the floor, I give it the potential to do work. Before, it could do nothing; it was just sitting there. But now, it has the ability, if released, to unleash work by transferring potential energy into kinetic energy. Remember the work-energy theorem from above? As the pencil falls to the ground, it will have a velocity, which gives it kinetic energy, which due to the work-energy theorem, allows it to do work.
Fig. 2 - At the top of the ramp, all the energy that the ball has is potential. Once it gets to the bottom of the ramp, all the ball's energy has been converted from potential to kinetic energy.
Potential Energy Formula
Ok, so I know what potential energy is, but how do I quantify it? That is what the potential energy formula is for. With the formula
$$\Delta U_\mathrm{g} = mg\Delta h\mathrm{,}$$
we can determine how much gravitational potential energy an object has.
Notice a few things.
- Potential energy is represented by a capital \(U\).
- The little \(\mathrm{g}\) subscript means that this formula is for gravitational potential energy only.
- \(m\) is the mass, \(g\) is the gravitational constant, and \(\Delta h\) is the change in height (so in the pencil example, it would be \(1\,\mathrm{meter}\) because the pencil was \(1\,\mathrm{meter}\) off the ground).
Conservative Force and Potential Energy
We couldn't talk about potential energy without talking about conservative forces. Don't worry; physics cares nothing for politics. We aren't about to get into a rant about the second amendment or anything. We will, however, keep the theme a little patriotic and talk about independence, well, path independence at least. (Yeah, I know your eyes probably aren't going to recover from rolling so much because of those terrible puns).
What is a Conservative Force?
In a nutshell, conservative forces are forces that are independent of path. This means that we don't care about the middle of the story. If I was a strict boss and wanted you to work at 9:00 a.m. sharp, I wouldn't care how you got there, just that you did get there. I don't care if you took the bus, car, train, or plane: as long as you're there at 9:00 a.m. That's how conservative forces work.
Fig. 3 - Notice how the path taken does not matter; either way, we're getting from \(E=0\) to \(E=mgh\).
Remember this: The work done by a conservative force will be zero if the object's path is closed. This means that the object's displacement would be zero (meaning that the object's final and initial position are the same).
It is also essential to make note of another type of force: dissipative forces.
Dissipative forces are forces where energy is not conserved.
Therefore, they are different than conservative forces in that respect. Examples of conservative forces are friction, resistive forces, or forces outside the system.
Relation Between Conservative Force and Potential Energy
Potential Energy allows for work to be done independent of path. Therefore, you can think of it like this: potential energy "sets up" the perfect scenario for conservative forces, giving them something of an alley-oop.
By understanding the relationship between conservative forces and potential energy, we can derive a formula for an object-earth system:
$$U_\mathrm{g} = \frac{-Gm_1 m_2}{r}\\\mathrm{.}$$
Notice here the similarities between this equation and the potential energy equation above.
Similarities | $$\Delta U_\mathrm{g} = mg\Delta h$$ | $$U_\mathrm{g} = \frac{-Gm_1 m_2}{r}\\$$ |
Gravitational Constant | $$g$$ | $$G$$ |
Mass | $$m$$ | $$m_1,\,m_2$$ |
Distance between objects or height | $$\Delta h$$ | $$r$$ |
Remember that at an infinite distance away from the earth, the potential energy of the object-earth system would be zero because as \(r\) gets really large, that fraction would approach zero.
Application of Conservative Force and Potential Energy
By understanding the relationship between conservative forces and potential energy, mathematicians and physicists devised this fancy mathematical relationship:
$$\Delta U = -\int_a^b \vec F_{\mathrm{cf}} \cdot \mathrm{d} \vec r\mathrm{.}$$
We could really go down the rabbit hole and go into the derivation of this formula, but I'll spare you the mathematical mess. This means that potential energy is equal to the integral of the conservative force times the displacement. Does that sound familiar? It sounds similar to our formula for work! It's almost like work and potential energy are related...wait, isn't energy just the capacity to do work?
If you were to differentiate that integral to figure out the force with respect to the potential energy, the formula
$$F_x = \frac{-\mathrm{d}U(x)}{\mathrm{d}x}\\$$
would result.
Note that this equation shows us that the conservative force equals the negative derivate of the potential energy.
Conservative Force and Potential Energy Examples
An ideal spring is an example of a conservative force, whose force is represented by the equation
$$\vec F_\mathrm{s} = -k\Delta \vec x\mathrm{.}$$
The force with the underscore \(s\) means the force of the spring, \(k\) is the spring constant, and \(x\) is the displacement of the spring.
The formula
$$U_\mathrm{s} = \frac{1}{2} k \Delta x^2$$
is used to calculate the potential energy in a spring.
Nonlinear forces exerted by a spring are also conservative in their own spring-object system.
Calculate the force of a spring pulled \(1\,\mathrm{m}\) from equilibrium whose spring constant is \(2\,\mathrm{\frac{N}{m}\\}\).
Use the formula
$$\vec F_\mathrm{s} = -k\Delta \vec x\mathrm{,}$$
which models the force exerted by a spring. Then, recall that
$$k=2\,\mathrm{\frac{N}{m}\\}$$
and
$$\Delta x = 1\,\mathrm{m}\mathrm{.}$$
Finally, use those numbers in our force equation:
$$\vec F_\mathrm{s} = -2\,\mathrm{\frac{N}{m}\\}(1\,\mathrm{m}) = -2\,\mathrm{N}\mathrm{.}$$
We've come a long way since the first of this article. You've just been overloaded with information; now it's clutch-time. We are on the green and it's time to put the ball in the hole. These are the most essential concepts you will need to remember.
Conservative Forces and Potential Energy - Key takeaways
Energy is the capacity to do work.
Work is equal to the change in kinetic energy.
Potential energy is energy inherent in an object due to its position relative to others or its intrinsic characteristics.
Conservative forces are forces that are independent of path. Dissipative forces are forces that do not conserve energy.
Gravitational forces and forces from an ideal spring are examples of conservative forces.
The work done by a conservative force will be zero if the object's path is closed.
The conservative force equals the negative derivate of the potential energy.
References
- Fig. 1 - (https://pixabay.com/photos/golfing-golfer-man-swinging-club-78257/) pixabay free license
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