Once a student came to see me. He did not do well on a Test which he had just taken. He and I went through some of his mistakes and explored ways which might help him to improve his future performance. I think it is beneficial for me to repeat some of our analyses here. For my discussion below, I will refer to the student as Joe. While the discussion below uses specific examples, please bear with me on the details. As you will see the message we will arrive at toward the end is quite general.

1. On the problem of rolling down the incline. Joe got a torque which is mgRcosq. He said that it was just "a careless silly mistake". There is a comonsense technique which may be helpful in checking one's answer. Consider extreme situations, and see if the the expression makes sense. For instance, here if we set theta=0, which corresponds to a flat surface, we should expect a zero acceleration, which contradicts the present answer. Here instead, the toque given by t=mgR.

2. Joe then proceeded to explain that since he made a silly mistake, he figured out the corresponding acceleration: a=(5/4)gcosq as the correct answer. He said that taking into account the cosine instead of the sine mistake, his improved answer would be a=(5/4)gsinq. This answer is still no good. To see this, consider the case, where the incline is vertical. Here we set q=90 deg. The so-called "improved answer" leads to a=5g/4. This is wrong. The acceleration should never exceed g!

3. We then turned our attention to the problem of two-satellites. Here the question concerns the effect due to the doubling of the radius. Since gravity has an inverse-square law, associated with the doubling of the radius, one should get a factor of 4. Joe gets a factor of two. He says that he understands what goes on. Unfortunately the very "silly mistake" takes us from the inverse second-power law to the inverse first-power law. The physical phenomena implied by these two power-laws are completely different.

I would like to impress all of you, that physics is different from mathematics, in that physics is a disicpline which describes and predicts what takes place in nature. In order for us to have some chance to understand nature, we need to use precise logical thinking. For example, the difference between "sine" and "cosine", "factor of two" etc must be taken seriously. They are not merely symbols on a paper, they do lead to different predictions in nature.

In order for students to do well in this class, they are required to do rigorous manipulation of symbols. Remember they represent what goes on in nature. Train yourself to be rigorous in algebraic manipulations.

Typically a Test question involves 2-3 steps of logical reasoning and algebraic manipulation. When one does not take each step too seriously, one cannot perform well. One gets very few questions right. Once one approaches each step and each question in a more systematic and serious manner, he/she will begin to perform.