We continue our study of objects in motion with a focus on Newtonian analysis (NGSS Performance Expectation HS-PS2-1). In addition to our work with forces, today I begin to link forces and energy, demonstrating how the conservation of energy (NGSS Performance Expectation HS-PS-3-1) can be found in virtually all of Physics.
Today's warmup problem features two interconnected masses, similar to the set-up of one of our lab stations from a few weeks back. Before students begin their work with this problem, I take a few minutes to elaborate on the idea that the tension in the rope that connects the masses is the same everywhere. It is not an obvious trait and students need to be reassured that it is true. Furthermore, though this is more obvious, I remind students that the two masses move together and, therefore, will experience the same acceleration, though in different directions. My intent is to minimize the number of distracting elements in what is surely a difficult problem.
Students work individually or in small groups for six or seven minutes while I visit with them to provide reinforcement and redirection, if necessary. After that time, I give students a two-minute warning, hoping to motivate them to wrap up their thinking on this problem before we discuss it at the board.
At the board, I lead a discussion about the solutions to these problems. I stress the creation of a free-body diagram and net-force equations before diving into any algebraic solution. As you can see in the first solution, I don't add in any of the known information until deep into the problem.
Once the tension in the rope is found, we can use that information for the second part of the problem. Again, I stress the creation of the free-body diagram and net-force equations before doing any substitutions of known quantities.
After solving these problems, I ask students if they are bothered by the fact that a smaller mass is getting a larger mass to move. It's a mixed response but the question does motivate some student conjectures. Some will say that gravity is '"not acting on m1" (the mass on the table) which, upon further probing, is withdrawn and re-stated. Eventually, we get to the truth; the motivating force in this scenario is the earth pulling on m2 while the hindering force is the frictional force on m1. The difference in those forces determines whether the objects will accelerate or not.
We end this segment of class and prepare for a one-question quiz.
As a way to obtain a quick formative assessment of my students' understanding, I use "one-question quizzes." These quizzes allow students to show their depth of knowledge on a particular topic with essentially zero risk to their grades. I allow each student to identify the lowest score (out of 20 points) that they would want to count. For example, a student who indicates an 18/20 would receive a grade of 18 (or higher) if her quiz work warranted such a score, but would be exempt from the quiz if she failed top score at least an 18. There is, therefore, an incentive to do well though no risk. Normally, I use these quizzes immediately after a practice session. Today, however, it follows the warmup problem, though we had a substantial practice session in the preceding class.
Students spread out throughout the room and take their notes with them to access during this one-question quiz. I have them alter two items on the first question of their practice set: the mass becomes 4.33 kilograms and the coefficient of friction changes to .225.
Students work on this problem individually. After about ten minutes, most students are done, though a few take just a bit more time. I ask those students who need more time to move to the back of the room - they are reassured that they can take whatever time they need, but the class moves forward while they are working.
Here's a successful effort where the process is shown clearly. Note the free-body diagram in the upper-left and the net-force equation that drives the algebraic solution to the right. Also note the "16/20" indicated in the upper right-hand corner; that is this student's lowest-acceptable score!
Here's a sample of the most common mistake made: mid-way through the problem this student forgot that the mass had changed. I still give plenty of credit for insight into the Newtonian process.
When the majority of students have completed their work, I ask them to take out their notebooks for some new thoughts about energy and motion.
In the final segment of class, I want to connect our year-long thread of energy conservation with objects in motion. Today, we start with an idealized arrangement - we assume a frictionless surface. Thought this is clearly incorrect, it helps us to get started.
I show a diagram on the board and ask students to take 30 seconds to think carefully about the prompts. Before soliciting any responses, I ask my students if they are confident about the energies at at least one of the three areas indicated (top of ramp, along the bottom stretch, and up the slight hill on the right). After seeing multiple hands go up, I ask a student to describe what's happening to the potential energy in any one of the three locations. I then call upon others to describe the potential energy in the other regions. It is surprising how complex the answers are given the fact that the potential energy depends, in this simple case, only on the changing height. Indeed, most of the discussion here is directed toward seeing that simple connection.
Having established the changes in the potential energy, I ask students to speculate on the variation in kinetic energy at the same locations. This is also surprising, though in a different way. Students are absolutely on target with their thoughts and they exploit the conservation of energy idea to express themselves. It is only now evident to me that their earlier struggles (describing the potential energies) were about limiting their thoughts to potential energy - they understood the conservation of energy and were using conservation arguments to describe the potential energies. In other words, I was looking for simple insights (that the potential energy of a mass changes with height) and they were providing complex insights (that potential energy will decrease while kinetic energy increases due to conservation ideas).
I ask students to compute the potential and kinetic energies at these three locations and they do so with great ease. I record the results at the top of the hill and at the end of the ramp on the board. based on their thoughts.
We end the day with a simple exercise. First, by raising or lowering our right hands, we qualitatively show the magnitude of the potential energy as the cart moves from beginning to end. Then, we repeat the process with our left hand for the kinetic energy. Finally, we use both hands to simulate the exchange of potential for kinetic energy. The alternating waving hands is a good formative assessment of student understanding of the conservation of energy - at least in this idealized case of no friction. Later in this unit we will add in the complexity of friction.