LECTURE 12: Free body diagrams Select LEARNING OBJECTIVES: i. ii. iii. iv. v. vi. Understand how to define a system for which to draw a FBD for. Demonstrate the ability to draw a properly scaled free body diagram from an image of a scenario or a written description of a scenario. Understand the importance of including a coordinate system along with a FBD. Be able to determine the relative direction of the acceleration of an object based off of a properly scaled FBD. Understand that the two forces constituting a force pair will never show up on the same FBD. Be able to identify any internal forces for a system if applicable. TEXTBOOK CHAPTERS: Giancoli (Physics Principles with Applications 7 th ) :: N/A Knight (College Physics : A strategic approach 3 rd ) :: 4.3, 4.6, 5.7 BoxSand :: Forces ( Free Body Diagrams ) WARM UP: After a baseball with mass m1 is hit, it flies through the air from the home plate (HP) to the outfield from left to right as represented by the trajectory drawn below. Ignoring air resistance, which of the following FBDs for the baseball at point P could be correct? To help us properly analyze the motion of a system, we need to consider all the interactions that the system has with its surroundings before we write down any equations. The number of interactions can become daunting, not to mention keeping track of the magnitudes and directions of all the forces that we use to mathematically describe these interactions. A free body diagram (FBD) is an invaluable tool to help us visualize all the forces acting on our system so that we may analyze the motion of the system much more efficiently. Until otherwise noted, we will still be working under the point particle approximation. Thus all forces we draw on objects can be represented as acting through the object's center of mass, which itself can be represented by a point particle. Do not concern yourself with the actual calculation of the center of mass statement just yet, we will introduce that math at a later time. For now it is sufficient to just understand that we are writing equations of motion for the center of mass of our system, wherever that may be located. Lectures Page 1
Defining a system The first step in analyzing any problem is deciding what object(s) we are interested in. This is known as defining our system. So the system is just another word for the collection of objects, or single object, that we plan on analyzing to determine motion. We restrict ourselves to only defining a system of multiple objects if all objects have the same acceleration. Consider the collection of objects below. Where mass 1 and 2 do not slide relative to one another (i.e. mass 2 will always be directly on top of mass 1), and mass 3 is connected by a massless string which does not stretch. Since mass 2 does not slide relative to mass 1, they both have the same acceleration to the right. Also, since mass 3 is connected to mass 1 via a string that does not stretch, mass 3 has the same acceleration as mass 1. Thus, as the person is pulling mass 3 to the right via a massless string, all objects have the same acceleration. With this image constructed, we now have options for how we can define our system. The options for our system are (m 1, m 2, m 3, m 1 + m 2, m 1 + m 3, m 2 + m 3, m 1 + m 2 + m 3 ). A few of these options are shown below. Drawing a FBD Now that you have defined your system, it is time to draw a FBD for that system. Our FBD will consist of a single dot, representing our system, from which we draw force vectors that represent all the interactions that the system has with its environment. FBD are not complete without a coordinate system defined. To illustrate what a FBD is, we will look at two examples. EXAMPLE: Draw a FBD for the defined system below. Assume no friction exists between m 1 and the surface, and also that m 2 is stuck on top of m 1. Lectures Page 2
Since this is not a video, I will show the important steps that I take when drawing a FBD. Showing all these steps are not necessary, they are only shown for pedagogical purposes. After this one example, I will only show the final free body diagram(s) without the steps. Step 1: Draw horizontal and vertical axes Step 2: Draw non-contact forces Step 3: Draw contact forces NOTE: I like to draw my non-contact forces first, just so I don't forget them (e.g. I think about all the non-contact forces that exist and see if any of them would be present on the FBD of interest). Then I go through the long list of contact forces in my head to make sure I don't forget any of them either. You do not need to draw them in this order, I just use it as a systematic way so that I hopefully don't forget to include any necessary forces. Lectures Page 3
Step 4: Draw/label coordinate system ** This is the final finished FBD with the forces scaled appropriately. EXAMPLE: Draw a FBD for the defined system below. Assume no friction exists between m 1 and the table, also no friction between m 3 and the table, and that m 2 is stuck on top of m 1. The steps shown in the first example before this one should be followed (i.e. draw horizontal and vertical axes, draw non-contact forces, draw contact forces, define coordinate system). PRACTICE: Draw a FBD for each mass separately and the combined (m 1 + m 2 ) system. Attempt to scale each force relative to each other. Lectures Page 4
PRACTICE: Draw a FBD for the person m 2, and the scale m 1. Attempt to scale each force relative to each other. What does a scale really measure? Scale PRACTICE: Draw a FBD for the box below. Attempt to scale each force relative to each other. Lectures Page 5
PRACTICE: Mass 1 and mass 2 are suspended by three separate wires of negligible mass. Draw a FBD for each separate mass. Attempt to scale the forces relative to each other. *Do you see another system we could construct?* Lectures Page 6
Conceptual questions for discussion 1) 2) Will a force-pair ever show up on the same FBD? Consider the image below. Is there a normal force from the person on mass 2? 3) Consider the image below. How many systems could you construct to draw FBDs? Lectures Page 7