Newton's Second Law (AQA A-Level Mathematics): Revision Notes

Newton's second law of motion is fundamental in mechanics, expressing the relationship between the force applied to an object, its mass, and the resulting acceleration. In vector notation, this law is expressed as:

$\mathbf{F} = m\mathbf{a}$

Where:

• $\mathbf{F}$ is the force vector (in Newtons, $\text{N}$ ).
• $m$ is the mass of the object (in kilogrammes, $\text{kg}$ ).
• $\mathbf{a}$ is the acceleration vector (in metres per second squared, $\text{m/s}^2$ ).

In 2D (Two Dimensions)

Let:

• $\mathbf{F} = F_x \hat{i} + F_y \hat{j}$
• $\mathbf{a} = a_x \hat{i} + a_y \hat{j}$ Then:

$F_x \hat{i} + F_y \hat{j} = m(a_x \hat{i} + a_y \hat{j})$

This equation can be split into two separate equations for each component:

• Along the x-axis: $F_x = m a_x$
• Along the y-axis: $F_y = m a_y$

In 3D (Three Dimensions)

Let:

• $\mathbf{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$
• $\mathbf{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$ Then:

$F_x \hat{i} + F_y \hat{j} + F_z \hat{k} = m(a_x \hat{i} + a_y \hat{j} + a_z \hat{k})$

This results in three separate equations:

• Along the $x$-axis: $F_x = m a_x$
• Along the $y$-axis: $F_y = m a_y$
• Along the $z$-axis: $F_z = m a_z$

📝Problem: A 5 kg object is subjected to two forces: $\mathbf{F_1} = 10 \ , \text{N} \ , \hat{i}$ and $\mathbf{F_2} = 20 \ , \text{N} \ , \hat{j}$ . Find the acceleration vector $\mathbf{a}$ of the object.

In vector notation, Newton's second law $\mathbf{F} = m\mathbf{a}$ highlights the directional nature of forces and accelerations. By breaking down vectors into their components along the axes, you can solve for unknown quantities in both 2D and 3D problems. This approach is essential for accurately analysing the motion of objects under the influence of multiple forces.