Mechanical Models & Units

Decide what the model includes

A mechanics model replaces a physical situation with objects, forces and equations. State the assumptions that make those equations valid, then interpret the solution in the original situation.

Interpret the formula in its setting

Does direction matter?

On a line, choose one direction as positive. A negative velocity then means motion in the opposite direction. Speed is the magnitude of velocity, so it cannot be negative. Zero speed is possible; zero is not positive.

Keep a consistent unit system

The SI base units used most often here are the metre ($\mathrm m$), kilogram ($\mathrm{kg}$) and second ($\mathrm s$). A derived unit combines base units using the relationship between quantities.

Convert each factor separately

Useful relationships are $1\,\mathrm{km}=1000\,\mathrm m$, $1\,\mathrm m=100\,\mathrm{cm}$ and $1\,\mathrm{cm}=10\,\mathrm{mm}$. Also, $1\,\mathrm h=60\,\mathrm{min}=3600\,\mathrm s$, $1\,\mathrm{kg}=1000\,\mathrm g$ and $1\,\mathrm g=1000\,\mathrm{mg}$. If a unit is squared or cubed, apply the same power to its conversion factor. For unfamiliar units, such as feet or miles per hour, use the conversion factor supplied in the question.

Work component by component

The perpendicular vectors $\mathbf i$ and $\mathbf j$ each have magnitude $1$ and point along the chosen positive coordinate axes. For a vector $\mathbf q$, $$\mathbf q=x\mathbf i+y\mathbf j=\begin{pmatrix}x\\y\end{pmatrix},\qquad |\mathbf q|=\sqrt{x^2+y^2}.$$ Printed vectors are usually bold; in handwriting, underlining a letter identifies it as a vector. Add or subtract matching components. Multiplication by a scalar changes both components; a negative scalar reverses the direction. A head-to-tail sketch of vector addition gives the same resultant as adding the components.

For equilibrium, the resultant must be the zero vector. A force $-\mathbf R$ cancels a resultant $\mathbf R$ component by component.

Show the component operations

Convert a magnitude and angle into components

Find a bearing outside the first quadrant

Identify the interaction

A force has magnitude and direction and is measured in newtons. Mass is a scalar measured in kilograms; it describes the object’s inertia. Weight is the gravitational force on that mass. The same object keeps its mass when moved to a place with a different $g$, but its weight $mg$ changes.

Connect the criticism to the prediction

A model is useful only within the conditions where its assumptions are reasonable. Compare its predictions with the physical situation. A refinement adds an effect that was omitted or represents it more accurately.