Dynamic equilibrium; equations of motion

Dynamic equilibrium

Newton's second law for a particle is ΣF = m a, which can be written as ΣF − m a = 0. If an imaginary inertia force −m a (equal in magnitude to mass × acceleration and opposite in direction to the acceleration) is added to the real forces, the body can be treated as being in equilibrium. This state is called dynamic equilibrium (D'Alembert's principle). The equations of statics can then be used to solve kinetics problems. Example: for a block pulled along a rough floor with acceleration a, P − F − m a = 0.

Equations of motion

Rectilinear motion (along x):

• ΣF = m a
• ΣFₓ = m ẍ = m dv/dt = m v dv/dx (The sum of the forces perpendicular to the path is zero.)

Curvilinear motion:

• Rectangular components: ΣFₓ = m aₓ = m ẍ, ΣF_y = m a_y = m ÿ, ΣF_z = m a_z = m z̈
• Tangential and normal components: ΣF_t = m a_t = m dv/dt; ΣF_n = m a_n = m v²/ρ
• Radial and transverse components (polar): ΣF_r = m(r̈ − rθ̇²); ΣF_θ = m(rθ̈ + 2ṙθ̇)

where ρ is the radius of curvature of the path.