Recently, I have written about the importance of frequent problem solving for physics students. Thomas Kuhn, the philosopher of science, wrote that:
Students of physics regularly report that they have read through a chapter of their text, understood it perfectly, but nonetheless had difficulty solving the problems at the end of the chapter. Almost invariably their difficulty is in setting up the appropriate equations, in relating the words and examples given in the text to the particular problems they are asked to solve. Ordinarily, also, those difficulties dissolve in the same way. The student discovers a way to see his problem as like a problem he has already encountered.
This post explores this idea, demonstrating the importance of two types of knowledge: subject knowledge and procedural knowledge in learning physics. I have used an example to demonstrate the knowledge involved.
This example is adapted from a lovely physics book: 200 Puzzling Physics Problems (with hints and solutions) by Gnadig, Honyek and Riley (2001). The puzzles (without the hints or solutions) can be found here.
A bottle of water is suspended from a fixed point by a inextensible rope. The bottle is set in motion and the system swings as a pendulum. However, the bottle leaks and the water slowly flows out of the bottom of it. How does the period of the swinging motion change as the water is lost?
There is a lot of knowledge hidden needed to solve this problem:
- A fixed point is a steady attachment, such as a firm hook in the ceiling.
- An inextensible, or non-elastic rope, string or cable is a mainstay in physics questions. Harder questions (or real situations) may take the elastic properties of the string into account. This is a training question – the question setter has taken one complication out of the situation. (Other mainstays include the massless rope and the frictionless pulley – included to simplify the problem).
- The question states that the bottle and water is a pendulum (we call this a “system”). The longer the pendulum, the longer the period of oscillation.
- The period of the pendulum depends on the length of the pendulum – the distance from the fixed point to the centre of mass. The total mass of the system makes no difference to the period of oscillation, but the position of the centre of mass does.
- The student has to realise that as the water leaks away, the centre of mass of the system starts to move which changes the length of the pendulum.
In addition to the physical knowledge, the student also needs to know several problem solving strategies, including the knowledge of how to represent the situation as a diagram – especially knowing that the bottle can be represented as a 2D rectangle, which you can see in my diagrams below.
All of that is background knowledge. It all has to be stored in the solver’s long term memory for low-effort recall. That means, the student will have solved many problems involving lengths of pendulums and finding centers of mass before.
My first thought in solving this problem was that as the water leaks, the centre of mass of the system will move downwards.
At first I thought this was the full solution, although my experience made me suspicious that this was too simple. So I checked using an extreme case. What happens when all of the water has leaked out?
My initial solution would put the centre of mass low in the bottle. But this can’t be right. When the bottle is empty, the centre of mass is in the centre of the bottle, just as it would be when the bottle is full.
Both the full and empty bottles have the centre of mass in the centre of the bottle.
So, I have three situations:
- The full bottle – the centre of mass is in the centre.
- The half full bottle – the centre of mass is lower.
- The empty bottle – the centre of mass is in the centre.
The solution is then, at first, the pendulum’s period increases as the centre of mass moves downwards, but then, at some point, the period starts to increase again until the bottle is empty and the period is back to where it started from.
There is an extension to this question: how full is the bottle when the period of oscillation is greatest? Solving the extension adds another fact to the schema – implication: always do the extension questions.
I have shown this problem and it’s solution to show the process to a non-specialist. Great physicists are not born, they are made, problem by problem. The memory of this problem and how to solve it can be recalled from long term memory when solving another problem involving pendulums or moving centres of mass as well as reinforcing the importance of considering the boundary conditions – the empty/full bottle. Finally, the solution adds experience – was the problem too easy? Have I missed something? Do I need to check? There is a sense of completeness you get from solving a problem which keeps you coming back for more. In the end, I believe it is the enjoyment students get from solving problems (like any exercise, it isn’t fun at first) that gets tem coming back for more and which turns them into new physicists.