Dynamics foundations – source of D’Alembert’s principle D’Alembert’s principle, which converts a dynamic problem into a static equivalent by introducing an inertial force, fundamentally depends on which Newtonian law?

Introduction / Context:
D’Alembert’s principle is widely used to analyze dynamics by recasting equations of motion into equilibrium form. It explicitly invokes an “inertia force” to balance applied forces, allowing the use of static tools like free-body diagrams with ΣF = 0.

Given Data / Assumptions:

• Newton’s second law: ΣF = m a for a particle.
• Inertial (non-accelerating) reference frame.
• Mass m is constant and motion is classical (non-relativistic).

Concept / Approach:
Rearrange Newton’s second law to ΣF − m a = 0 and interpret −m a as a fictitious inertia force acting on the particle. This converts a dynamics equation into a statics-like equilibrium condition, which is the essence of D’Alembert’s principle.

Step-by-Step Solution:

Verification / Alternative check:
For rigid bodies, an analogous step yields ΣF − m a_G = 0 and ΣM_G − I_G α = 0, again mimicking statics by adding suitable inertia forces and moments, still grounded in Newton’s second law.

Why Other Options Are Wrong:

• Incorrect / depends on frame: while inertial frames are required, the underlying basis remains Newton’s second law.
• Only valid for constant velocity: trivial case a = 0 reduces to statics; D’Alembert’s utility is precisely when a ≠ 0.

Common Pitfalls:
Treating inertia force as a real applied force; it is a mathematical device to achieve equilibrium form, not a physical interaction force.

Final Answer:
Correct