Dynamics

Dynamics explains why motion happens. Learn Newton's laws, F = ma, work–energy and power, impulse–momentum, rotational dynamics T = Iα, and how they size motors, brakes and lifting systems.

Dynamics links forces to motion. With Newton's second law you can find the force a motor must provide, the load a brake absorbs, or the stress in a crane cable during acceleration.

Three equivalent tools cover almost every problem: force–acceleration (F = ma), work–energy (useful when speeds and distances are known) and impulse–momentum (useful for impacts and short events).

An analogy

Pushing an empty shopping trolley versus a full one: the same push gives the full trolley much less acceleration. That resistance to changing motion is mass, and F = ma quantifies it.

Newton's three laws

1: a body stays at rest or constant velocity unless a net force acts. 2: net force = mass × acceleration. 3: every action has an equal and opposite reaction. Most engineering dynamics is the second law applied carefully with a free-body diagram.

Work, energy and power

Work = force × distance moved in its direction. Kinetic energy ½mv² and potential energy mgh convert into each other; friction turns energy into heat. Power is the rate of doing work — motors and engines are rated by it.

Impulse and momentum

Force × time equals change of momentum. Airbags and crumple zones extend collision time to reduce peak force. In collisions momentum is conserved; the coefficient of restitution describes how much kinetic energy survives.

Rotational dynamics and D'Alembert

For rotation, torque = I·α. A motor driving a load through gears sees the load inertia reflected by the square of the gear ratio. D'Alembert's principle treats −ma as an inertia force so dynamics can be solved like statics — the basis of dynamic force analysis of engines and mechanisms.

Formulas

• Newton's second law: ΣF = m · a
• Work–energy: ΣW = ΔKE = ½m(v² − u²)
• Power: P = F · v = T · ω
• Impulse–momentum: F · Δt = m(v − u)
• Rotation: ΣT = I · α, KE_rot = ½ I ω²

Worked example: Tractive force

Given: m = 1500 kg; a = 2 m/s²; Resistance 400 N

Find: Driving force.

• ΣF = ma → F − 400 = 1500 × 2
• F = 3400 N

Answer: 3.4 kN

Worked example: Energy to stop a vehicle

Given: m = 1200 kg at 25 m/s; Braking force 6 kN

Find: Energy absorbed and stopping distance.

• KE = ½ × 1200 × 25² = 375 kJ
• d = KE/F = 375 000/6000 = 62.5 m

Answer: 375 kJ, 62.5 m

Common mistakes

• Wrong: Forgetting weight when accelerating vertically. — Right: Lifting: T = m(g + a); lowering with deceleration also increases tension.
• Wrong: Adding kinetic energy of rotating parts only for translation. — Right: Include ½Iω² of wheels, flywheels and motor rotors in energy balances.

Applications

• Motor & drive sizing: Peak torque comes from acceleration of the reflected inertia; continuous torque from friction and load.
• Vehicle safety: Crash structures are designed with impulse–momentum to keep occupant deceleration survivable.
• Elevators & cranes: Rope tension and motor power are computed for acceleration, steady hoisting and emergency stops.

Key takeaways

• ΣF = ma with a complete FBD.
• Energy: KE = ½mv², P = Fv = Tω.
• Impulse = change in momentum.
• Rotation: T = Iα; include rotating inertia in energy.

Going further

• Lagrangian mechanics: For multi-body systems (robots, vehicles), Lagrange's equations from kinetic and potential energy produce the equations of motion systematically.
• Vibration: Mass + stiffness gives a natural frequency ω_n = √(k/m); dynamic loads near it cause resonance — why machines are balanced and isolated.
• Multibody simulation: Tools such as Adams or Simscape solve rigid/flexible body dynamics with contacts and joints for suspension, landing gear and robot design.