In today's lecture, we discussed various problem-solving techniques in physics.

Idea

Dimensional analysis: both sides of a valid equation must have the same dimensions.

The argument of an exponential, logarithm, or trigonometric function must be dimensionless.
Dimensional analysis can always be used to check your answers, and sometimes to solve a problem entirely up to a dimensionless factor.
Optional: Dimensional analysis can be used to solve integrals.
The fundamental dimensions are M (mass), L (length), T (time), K (temperature), A (current).

Tip: Estimation problems

When a problem asks you to ``estimate" an answer or to the ``correct order of magnitude" (power of 10), you can often ignore numerical prefactors in your answer to make calculations simpler.

Idea

Error analysis formulas. $A$ and $B$ are variables, and $c$ is a constant. The relative error $\delta_{rel}(A)$ is $(\delta A)/A$. \begin{gather} \delta(A+c)=\delta(A)
\delta_{rel}(c\cdot A)=\delta_{rel}(A)
\delta(A+B)=\sqrt{(\delta(A))^2+(\delta(B))^2}
\delta_{rel}(A\cdot B)=\sqrt{(\delta_{rel}(A))^2+(\delta_{rel}(B))^2}
\delta_{rel}(A^n)=|n|\cdot \delta_{rel}(A)
N \text{ trials } \rightarrow \text{ error reduced by } \sqrt N \end{gather}

There is always at least one error analysis problem on the $F=ma$, and the USAPhO may require some error analysis. An extensive use of error analysis is tested during physics camp.

Idea

Limiting cases: check limits to determine the validity of your answer. Does it (physically) make logical sense? If variables are brought to the limits $\lim_{x\rightarrow 0}$ and $\lim_{x\rightarrow \infty}$, does the answer make sense?