F = ma & Vector Notation

Column Vector Notation

• Vectors can be written as a vertical stack of components: $\begin{pmatrix} F_x \\ F_y \end{pmatrix} = m \begin{pmatrix} a_x \\ a_y \end{pmatrix}$. This format is often preferred for ease of addition and subtraction of multiple forces.

Unit Vector (i-j) Notation

• The unit vectors $\mathbf{i}$ and $\mathbf{j}$ represent one unit in the positive x and y directions respectively. The law is written as $F_x\mathbf{i} + F_y\mathbf{j} = m(a_x\mathbf{i} + a_y\mathbf{j})$.

• To find the magnitude of the resultant force from its components, use the Pythagorean theorem: $|\mathbf{F}| = \sqrt{F_x^2 + F_y^2}$.

• Step 1: Component Separation: Always treat the horizontal ($\mathbf{i}$) and vertical ($\mathbf{j}$) directions as independent systems. A force in the $\mathbf{i}$ direction has no effect on acceleration in the $\mathbf{j}$ direction.

• Step 2: Summing Forces: Calculate the resultant force vector by summing all individual force vectors acting on the body: $\sum \mathbf{F} = \mathbf{F}_1 + \mathbf{F}_2 + ... + \mathbf{F}_n$.

• Step 3: Apply N2L: Use the equation $\mathbf{F}_{net} = m\mathbf{a}$ to solve for the unknown vector (either force or acceleration).

• Step 4: Recomposition: If the question asks for magnitude or bearing, convert the final vector components back into a scalar value and an angle using trigonometry.

• Weight as a Vector: While mass is a scalar, weight is a force vector that always acts vertically downwards. In a 2D vertical plane, it is represented as $\mathbf{W} = -mg\mathbf{j}$.

• Check Units: Ensure mass is in kilograms (kg) before applying $F=ma$. If given in tonnes, multiply by $1000$.

• Directional Consistency: Always define a positive direction for $\mathbf{i}$ and $\mathbf{j}$ at the start. Resistive forces like friction or air resistance should usually be negative if they oppose the direction of motion.

• Resultant vs. Individual: Examiners often provide multiple forces. You must find the resultant (the sum) before setting it equal to $m\mathbf{a}$.

• Sanity Check: If an object is moving at a constant velocity, the resultant force vector must be $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ or $0\mathbf{i} + 0\mathbf{j}$.

• Mixing Components: A common error is adding an $\mathbf{i}$ component to a $\mathbf{j}$ component. These must remain separate throughout the calculation until the final magnitude is found.

• Weight Direction: In 'bird's-eye view' problems (horizontal plane), weight acts into the ground and is often ignored unless friction is involved. In vertical problems, weight must be included as a downward force.

• Squaring Negatives: When calculating magnitude $|\mathbf{a}| = \sqrt{a_x^2 + a_y^2}$, remember that squaring a negative component results in a positive value.