Today's warmup problem is actually two problems - one that checks for understanding about the nature of forces as vectors and one that requires some computation. The consideration of multiple forces and how they combine to create a "net force" is a critical step in the development of Newton's Second Law of Motion (NGSS Performance Expectation HS-PS2-1). Our recent work has established the connection between force, mass, and acceleration. Today's work enriches that understanding by providing students with the ability to calculate two common forces: friction and weight.
Students work individually or with one another as I wander through the room to provide targeted feedback. I look for the level of understanding of each student and have conversations to stimulate student thinking. Though today students are working on two problems, the first one is relatively quick and depends only on recognizing the number of forces that cancel one another. I try to get students to the second problem as quickly as possible. The second problem is multi-faceted and takes students several minutes to consider. Students work on these problems for 12 to 15 minutes.
I start the discussion of the solutions with the intent of developing a "protocol" for solving Newtonian problems - free-body diagrams and net force equations should precede any attempts to solve for variables. In addition, I want to review some earlier concepts and show how they fit in nicely with these kinds of problems. During this segment, I work through each problem at the board while students follow along and check their work with mine.
For example, the first warmup problem is just a chance to probe student understanding about forces as vectors - numbers with a sense of direction. The problem at first glance looks complex but becomes extremely simple if one understands that the forces pointing in the cardinal directions (the red ones) all cancel one another leaving only the fifth force (in blue) remaining. I take a moment to remind students that we have dealt with vectors for quite a while and have even done vector addition with forces back in our Electrostatics unit! Furthermore, I remind them about the definition of vectors; numbers with magnitudes and directions.
The second warmup problem has multiple parts and I address each one in turn. The first part allows me to show a free-body diagram and net force equations that lead to the determination of acceleration. This, in turn, allows me to connect back to our study of kinematics just recently - the acceleration allows us the chance to predict a future velocity.
The second part affords another opportunity for me to show the diagram/net force equation combination again. Students struggle with the notion of the "wall pushing" as it gives the wall too much of a sense of action. I appeal to their experience of walls being resistant to motion; this seems to help many add in a force from the wall acting on the cabinet.
The final part - involving the worker and his fictionalized sister - gives students one more chance to see how to proceed. We must define the forces and give them names in order for our drawings to have any real meaning. For students used to being provided all parts of a problem, this is a novel and daunting task. At the end of this sequence, they have seen three examples of that kind of creative problem-solving. The mathematics is relatively simple; the active definition of forces is the hard part!
We continue in the lecture mode for another few minutes. I introduce the idea that some forces can be calculated, noting that, in every problem to date, we have been provided directly with values for forces. I handout a single sheet of notes and have students follow along as I elaborate on them.
The two forces I introduce are the gravitational force (or the "weight") and the frictional force. I draw the distinction between "mass" and "weight" and note that, wherever "g" is different (in space, on the Moon, other planets), one's weight would change even as mass is constant.
For the frictional force, I choose to introduce it as the portion of an object's weight that gets converted into a force opposing motion. I tell students that this is a temporary, but useful, definition which allows us to start an earnest practice of Newtonian problems quickly. Knowing that the problem set I have prepared has only objects that lay flat on tables and floors, I know that this definition is sufficient for today's purposes. Indeed, at the end of class, I ask students to consider a problem with this definition (see "Exit Ticket"). So, though not ultimately fully truthful, it is a simple way to get students thinking about, and working with, the idea of friction.
Having shared these ideas with students, I provide them with a practice set and some time to develop their understanding.
I distribute the friction practice set and encourage students to attempt free-body diagrams before moving too quickly to solutions. Students are welcome to collaborate with one another or work individually. I circulate around the room looking to provide very targeted feedback.
In this image, the student has heeded my advice and has begun her solution with a free-body diagram. This is a bit unusual for the first practice set; many students don't see the value of this step and believe that, as a teacher, I'm just being picky!
In the image below, this student has done well until part c of the first problem where, after identifying the frictional force as 33.5 newtons, she has inserted some other force in the place where friction should have been. A conversation right now provides her with an appropriate re-direction.
Students continue with the practice set until there are about five minutes left in class. At that time, I quickly get their attention at the board and ask them to record their thought in response to an "exit ticket."
I the final few minutes, I distribute some small scraps of paper for students to respond to an exit ticket about friction. Here, I pose a problem for which our temporary definition of friction is inadequate. I read the problem out loud while showing it at the board and ask students to put down their best thoughts on the scrap of paper, without their names.
As an example, this student has tried to identify the places where he thinks friction is acting and, with his free-body diagram, has indicated how gravity is acting on the object ("Fw"). Though no sentences, this student has provided me with a quick mental model.
I collect these from each student and use them to get a better sense of their understanding of friction. This helps me set up my next lesson.