3.2 Dynamics

Picking the Right Method

Dynamics is worth 4–6 FE Civil questions. The exam rewards method selection: use kinematics when motion is asked, Newton's second law when forces and acceleration link, work–energy when speed relates to distance, and impulse–momentum when speed relates to time or collisions.

Kinematics

For rectilinear motion with constant acceleration a:

If acceleration varies, integrate: v = ∫a dt and s = ∫v dt. For curvilinear motion, use normal/tangential components: tangential aₜ = dv/dt and normal aₙ = v²/ρ, where ρ is the radius of curvature.

Projectile Motion

Horizontal velocity is constant; vertical motion has a = −g (9.81 m/s² or 32.2 ft/s²). On level ground:

$R = \frac{v_0^2 \sin 2\theta}{g}$

Maximum range occurs at θ = 45°. Time of flight is t = 2v₀sinθ/g.

Newton's Laws

Newton's second law for a particle is ΣF = ma. Build an FBD exactly as in statics, but the net force is no longer zero — it equals mass times acceleration. For connected bodies (pulleys, blocks), write ΣF = ma for each mass and link them through a common acceleration.

For rigid-body plane motion: ΣF = ma_G at the mass center and ΣM_G = I_G α for rotation, where I_G is the mass moment of inertia and α is angular acceleration.

Work and Energy

The work–energy principle avoids solving for acceleration:

$W_{net} = \Delta KE = \tfrac{1}{2}m v_2^2 - \tfrac{1}{2}m v_1^2$

• Work of a constant force: W = F·d·cosθ
• Work of gravity: W = −mgΔh (potential energy change)
• Work of a spring: W = ½k(x₁² − x₂²)

When only conservative forces act, mechanical energy is conserved: KE₁ + PE₁ = KE₂ + PE₂. Friction removes energy as heat.

Impulse and Momentum

Linear impulse–momentum links force and time:

$\sum F \, \Delta t = m v_2 - m v_1$

When the net external impulse is zero, linear momentum is conserved — the workhorse for collision problems. For a perfectly elastic collision, kinetic energy is also conserved; for a perfectly plastic (inelastic) collision the bodies move together afterward. The coefficient of restitution e relates separation to approach velocity, with e = 1 (elastic) and e = 0 (plastic).

Vibrations Basics

An undamped simple harmonic oscillator (mass m, spring k) has natural angular frequency:

$\omega_n = \sqrt{\frac{k}{m}}, \qquad f = \frac{\omega_n}{2\pi}, \qquad T = \frac{1}{f}$

A simple pendulum of length L has ωₙ = √(g/L). These free-vibration relations appear in the Dynamics section of the NCEES FE Reference Handbook — locate them before exam day so you only verify, not derive.