Expand Your Toolbox: Physics I and Beyond
An in depth look at the “rolling disk on an inclined plane” problem
There is often a point in your physics journey when you learn another way to work a problem after weeks, months, or even years of solving it one way. It typically occurs when someone shows you the new method and you say something like, “Well that is a cool way to look at it!”. I personally enjoy seeing these alternative methods as it is a fantastic opportunity to expand your toolbox.
In this article I will share with you three methods for solving for the acceleration of a rolling disk on an inclined plane without slipping. These methods span from techniques learned in introductory physics to graduate level classical mechanics.
Method 1: Energy Method
The first method we will introduce ranks at the most basic level. Using a combination of the conservation of energy and kinematics we can solve for the acceleration of the disk. Perhaps some readers will even recall solving this problem in their high-school or undergraduate physics course using this exact technique.
The energy method for solving this problem only requires knowledge of algebra and typically every equation you need would be provided by the instructor. Most physics instructors introduce the tools needed for this method in the first half of the course. I believe it is a necessity for all physics students to be able to use this method as it is a perfect example of the power of the conservation of energy and kinematics. Now let us expand your toolbox and take a look at Method 2, which is a much faster way of solving this problem.
Method 2: Torque Method
The torque method tends to be a quicker way to solve the problem. We will start with Newton’s 2nd Law of Motion.
As promised, Method 2 is faster and takes up far less page space than Method 1. Most students are introduced to Newton’s second law and torque near the middle of an introductory physics course and should become comfortable with solving the problem with this method. It is always fascinating to me that we can start a problem by applying different ‘basic’ laws of physics and get to the same answer. That just shows how powerful these techniques can be! Now let us expand our toolbox with a more complex method, amateurs be warned.
Method 3: The Lagrange Method
This final method requires some extensive knowledge of higher mathematics, including up through differential equations. Most folks learn this in an undergraduate mechanics series or a graduate level classical mechanics course. The Lagrange Equation method is a powerful formulation of mechanics that can be methodically applied to many problems with varying complexity. However, as the complexity of the problem increases, so does the difficulty in solving the equations of motion. I typically use software such as Mathematica to numerically solve the equations of motion for complex mechanics problems. Due to the simplicity of this problem, we can get to our solution analytically.
Our first step will be to refer to our visualization of the problem (I must stress the importance of always drawing a picture). We must define new variables x, which will be the distance the disk travels from the top of the inclined plane, and y to be the height of the disk. All distances measured to the center of the disk.
Now, we must form our Lagrange equation: L = KE - PE. Like in our first method, we need to define the kinetic and potential energies. The kinetic energy consists of two parts, the rotational kinetic energy and the translational kinetic energy. From there we can find our solution.
Some of you may wonder why anyone would use the final method to solve these problems when it takes so much time and page space. My argument is that the Lagrange formulation is a consistent methodology for solving these problems no matter the complexity. When you reach graduate level mechanics, you can have systems that are too complex for the first two methods. That being stated, the hardest part of the Lagrange equation is the set up. Once you get KE and PE you can use a computer to get to your final solution.
One of the main advantages of being a physicist is having the ability to look at a problem from 100 different perspectives. It is so very important to develop a toolbox of techniques and apply them to as many problems as you can. As you gain new knowledge, please pass it on. There is a significant shortage of physics educators and I encourage each reader to teach someone one of the techniques presented here. Spread the knowledge and expand someone’s toolbox!