Deep Dive: Second Law

a. Valid Only in Inertial Frames

In a non-inertial frame (e.g., a rotating or linearly accelerating frame), the relation $\vec{F}_{\text{net}} = m\vec{a}$ fails unless pseudo-forces (e.g., centrifugal, Coriolis) are added.

b. Consistency of Quantities

• $\vec{F}_{\text{net}}$, $m$, and $\vec{a}$ all refer to the same object.
• $\vec{F}_{\text{net}}$ is the vector sum of all external forces acting on that object.

c. Vector Nature

• $\vec{F}_{\text{net}}$ and $\vec{a}$ are parallel vectors: they point in the same direction, with magnitude related by mass.

d. Component Form

Newton’s Second Law applies independently along each coordinate axis: $$ F_x = m a_x, \quad F_y = m a_y, \quad F_z = m a_z $$

• Forces in perpendicular directions superpose linearly; the total acceleration is the vector sum of contributions from all forces.