Unit 4 Instructional Goals:
- Newton's 1st Law (Galileo's thought experiment)
- Develop the notion that a force is required to change velocity, not to produce motion.
- Constant velocity does not require an explanation.
- Force concept
- View force as an interaction between an agent and an object.
- Choose system to include objects, not agents.
- Express Newton's 3rd law in terms of paired forces (agent - object notation)
- Force diagrams
- Correctly represent forces as vectors originating on an object (point particle).
- Use the superposition principle to show that the net force is the vector sum of forces.
- A zero net force produces the same effect as no force acting on an object.
- Decomposition of vectors into components.
Unit 5 Instructional Goals:
- Newton's 2nd Law
- Develop mathematical models from graphs of acceleration vs force and acceleration vs mass.
- Introduce joint variation.
- CDP's dynamical properties, force diagrams and motion maps
- 2nd half of Newton's modeling cycle: deduce motions from forces
- Relate the directions of the acceleration and the net force vectors.
- Resolve vectors into components.
- Develop frictional force law
- Distinguish between static and kinetic friction.
First, it's really important to recognize there are a lot of preconceptions about forces. Things like air pressure causes gravity, a push or a kick imparts an impulse that gradually wears off, inanimate objects can't exert forces (they're just in the way), bigger objects exert more force during a collision, etc. It kills me that I haven't been asking to know what my students were thinking.
We started the unit with a quick pre-assessment to see what ideas were out there. The quiz itself was just a starting point. Afterwards, we had a "debate" where you would stand on a side of the room depending on whether you answered true or false to a specific question. Then, it was each sides job to convince the other side to join them. It's a fantastic way to see how students reached their answers. (I'm not sure how well it will work in a very small class, but I can see trying it with a class of 10 or 12.) Nothing will be resolved here, and it will be very important to keep a poker face while they're debating. No giving away answers! It'll make the exploration less fun.
Next, Laura led a discussion on the rules for forces, starting with the intuitive definition of a push or pull. She asked us what forces were acting on her when she was just standing still on the floor. Gravity was the first force mentioned, and she asked us what we knew about gravity. There were a couple of big ideas here. First, you don't have to be in contact with the ground for gravity to act on you. That led us to the idea of non-contact forces, which wasn't all that unusual, since we were all familiar with magnetism. It's just a matter of making it clear that there are non-contact forces, and that one of them is gravity. Second, that the Earth is what is doing the pulling. Gravity isn't an object - it's what we call the pull of the Earth. And here we got our essential rule for forces: something must be doing the pulling or pushing (exerting the force). This would be an ongoing theme, and a really useful rule later on. (If you think there's a force, but you can't figure out what's pulling or pushing, you have to call into question that there is a force acting.)
We also discussed other forces that could be acting, like air pressure. Eventually, we agreed that it was possible for the air molecules to be hitting her and exerting forces on her, but that they were so small we could ignore them. Finally, we moved to the question of whether the floor pushes up on her. This is where some students will have trouble accepting that an inanimate object can push. So we started by looking at pair of large spring. Laura pushed down hard on the springs, and we agreed they were pushing back up on her, because they wanted to go back to their normal shape. She put a book on top of the springs, and we agreed that because they were still squished (though not as much), they would push back up. She moved on to a sponge material, putting less and less mass on top, so that it was compressed less and less. We still agreed that if it was compressed, it would exert a force by trying to go back to its original shape. Next, she moved to a more rigid material - a white board. We could see it sagged when she put a book on it, so we knew that it was also exerting a force on the book. All of this was leading to the idea that all objects change their shape (including the floor), it might just be microscopically. (There is a way to do this on a table with a mounted laser and a mirror, but it sounds difficult to do...) I really liked the progression - it's so much more effective to see it than to just hear that the floor compresses microscopically.
The last thing we needed to figure out was which of the forces acting on Laura was the largest. She and Clint got in a pushing war on a chair, and we got to figure out what would make the chair move or not move. We had a rudimentary sense of balanced forces having no effect on the movement of an object, while unbalanced forces caused it to move. This, in turn, led us to believe the floor was pushing up on Laura just as hard as the Earth was pulling down. At this point, we established the rules for force diagrams, and drew one to represent Laura standing on the floor.
That took care of introducing the static case. Next was to get a feeling for what happens with moving objects. So, each group was given a mallet and a bowling ball. We had to figure out how to do 5 things with the bowling ball:
- Make it speed up
- After the ball is moving, keep it going at a constant speed
- After the ball is moving, make it slow down
- After the ball is moving, make it do a 90 degree turn
- Make the ball move in a circle
This was a great way to feel what was happening. Bryan asked us to draw the force diagrams for the bowling ball when it was not moving, speeding up, moving at a constant speed and then slowing down. We had some disagreement on whether a force was needed to keep the bowling ball moving forward at a constant speed. This is where we started to see the similarity of staying still and moving at a constant speed, and that was uncomfortable for some of us (in student mode, at least).
From here, we created some working definitions. Which, we then updated after Laura led us through some questions about what happens when we have two strings attached to a cart with equal or unequal masses at the other ends of them.
This is about where I would consider us to be in model deployment. At this point, we have an understanding of how forces cause objects to accelerate (not move). There are a lot more details to be added to this, but I think that's part of the refining during the deployment phase.
We did a quick introduction to forces as vectors (meter stick and light source to cast shadows on the wall and floor), and briefly talked about friction being a balancing force when an object is moving at a constant speed while being pushed. We also did an investigation between the force of gravity on an object and its mass to find a relation between them. This led us to the equation for weight. At this point, we were ready to work on some worksheets and discuss our work.
To introduce forces acting on an object on a ramp, we were asked to draw a free-body diagram of a cart moving down a ramp. I like this exercise, because it brings back the "what is doing the pushing or pulling" question. It's tempting to draw a force pointing down the ramp, because that is the direction of the acceleration. Answering the question about what is exerting that force leads to interesting discussions about the direction of the pull of gravity and the upward push of the ramp. Once we had settled on the force diagram, we talked about how to turn our coordinate axes so that gravity was the force that was at an angle.
At this point, we still only had a nebulous idea of friction. We explored it more by pulling a bowling ball (in a net) down a hallway at constant speed and measuring the force used to do so. Then we increased the speed to see if the force needed also increased. (The hard part is making sure the angle of pull is constant. Or else you have to find a way to pull perfectly horizontally - maybe a mechanic's cart?)
We also brainstormed some ideas about what factors affect friction. Then different groups tested different ideas. This might be difficult in my small classes - we'll have to spend several more days investigating, because we won't have as many groups to split the ideas between. In the end, though, we connected the changes we made that had any effect to the normal force or some property of the surfaces.
At this point, we were ready to (finally) develop Newton's 2nd Law. Using many different ways to create a constant, sustained force, we investigated what happened when we increased the force pulling on a cart with constant mass, and what happened when we pushed with a constant force on a cart with increasing mass. Now, our data for this lab wasn't great. It's difficult to produce a constant, sustained force by hand. But we did get some data, and were able to draw some conclusions in the group discussion. Mainly that we knew that acceleration was directly proportional to the force and that it was inversely related to the mass. By looking at the units for the coefficients in each of the two equations, we were able to conclude there was a joint variation: a = net F / m.
There were more worksheet problems to do and discuss once we developed Newton's 2nd Law.
The last thing we needed to do was develop Newton's 3rd Law, which is generally the hardest to understand. We started by measuring the force at each end of a string that was being pulled. We also checked what the force in the middle of the rope was, too. Then we used force sensors on carts on a track to measure the force of their crashes. We investigated all sorts of scenarios, and in each one, the forces between the carts were equal. (See how the spikes above and below the axis are at the same height? And at the same time?) This is where agent - object notation became important to differentiate between equal and opposite forces that do not form a 3rd Law pair and equal and opposite forces that do.
We didn't have a practicum at the end of these two units. We did go on to explore what happens in projectile motion, which is a combination of all the models we've built so far. The motion of a projectile is described by both the constant velocity model and the uniform acceleration model, and this is supported by examining the constant forces acting on a projectile in the air.