CELE Engineering Mechanics — Dynamics: KineticsSummary

The fundamental equation of kinetics states that the net force acting on a body equals its mass multiplied by its acceleration. This is a vector equation—forces and acceleration are directed quantities. In problems, you identify the body, draw a free-body diagram (FBD), resolve forces along the direction of motion, and sum them. The result gives either the acceleration directly (if all forces are known) or an unknown force (if acceleration is known). On inclines, resolve along and perpendicular to the surface; on horizontal surfaces with friction, include F_friction = μN opposing motion. This is the foundation of kinetics and should be your first approach when given forces and needing acceleration (or vice versa).

D'Alembert's principle reformulates ΣF = ma as ΣF – ma = 0, introducing an 'inertia force' (–ma) that acts opposite to the acceleration. This trick transforms a dynamics problem into an apparent equilibrium problem, allowing the use of statics methods (ΣF = 0, ΣM = 0). While mathematically equivalent to Newton's second law, it is particularly useful for problems involving multiple bodies with constraints (e.g., connected masses on pulleys or inclines) because it converts the system into a single equilibrium analysis. On the PRC exam, recognizing when D'Alembert is faster than direct application of ΣF = ma can save valuable time.

Work is the product of a force and the displacement in the direction of that force: U = F d cos(θ). The work–energy principle states that the net work done on a body equals its change in kinetic energy: U_net = ΔKE = (1/2)m v₂² – (1/2)m v₁². This principle is most powerful when the unknown is a velocity (speed) and distance is known, but time is not given. Key points: (1) Work of a constant force: U = F·d·cos(θ). (2) Work of gravity: U = ±Wh = ±mgh (positive when moving downward). (3) Work of a spring: U = (1/2)k(x₁² – x₂²)—this equals the negative change in elastic potential energy. (4) Friction always does negative work: U_f = –μN·d. When only conservative forces (gravity, springs) act, total mechanical energy is conserved: KE₁ + PE₁ = KE₂ + PE₂. This method is preferred over ΣF = ma when solving for final speed over a distance.

Impulse is the product of force and the time interval over which it acts: Impulse = F·Δt. Momentum is the product of mass and velocity: p = m·v. The principle states that the net impulse equals the change in momentum: ΣF·Δt = m v₂ – m v₁. This is the tool of choice when force and time are involved or when they must be found. Linear momentum is conserved in a system if no external impulses act on it, making it the foundation of collision analysis. Unlike energy (which can be lost in inelastic collisions), momentum is always conserved in collisions because the internal forces between colliding bodies are equal and opposite (Newton's third law) and act for the same time. This method is indispensable for impact, blast, and traffic accident analysis.

In a collision, momentum is always conserved, but energy may not be. The coefficient of restitution (0 ≤ e ≤ 1) quantifies the 'bounciness' of a collision and is defined as the ratio of relative velocity of separation to relative velocity of approach: e = (v₂ – v₁)/(u₁ – u₂), where u₁, u₂ are velocities before collision and v₁, v₂ are after. All velocities must be signed (positive right, negative left, etc.); never substitute bare speeds. For two bodies colliding, you have two equations: (1) momentum conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, and (2) restitution: e = (v₂ – v₁)/(u₁ – u₂). Solving these simultaneously gives final velocities. Special cases: e = 1 (perfectly elastic—KE conserved), e = 0 (perfectly plastic—bodies stick together), 0 < e < 1 (partially elastic—some energy lost). Sign errors are the number-one pitfall here.

When a body moves along a curved path, its acceleration has two components: tangential (a_t = dv/dt, along the path) and normal (a_n = v²/r, perpendicular to the path, toward the center). Newton's second law becomes ΣF_t = m a_t (tangential equation) and ΣF_n = m v²/r (normal equation). The normal equation is crucial for problems involving circular motion, banking of roads, and looping. For example, on a banked curve, gravity and normal force have components that together provide the centripetal force; on a vertical loop, the normal force must overcome both gravity and provide centripetal acceleration. The tangential equation governs the rate of speed change (speeding up or slowing down along the curve), while the normal equation governs the change in direction. Many exam problems disguise curvilinear motion in practical contexts (vehicle on a curve, object swinging on a rope, liquid in a rotating container).

The art of kinetics is knowing which principle to use. Use ΣF = ma when: (1) forces and acceleration are the primary variables, (2) time is not involved, and (3) the motion is along a fixed direction or path. Use work–energy when: (1) speed (kinetic energy) is the unknown, (2) distance is given, (3) time is not involved or not asked for, and (4) forces are conservative or friction is present. Use impulse–momentum when: (1) time and force are involved, (2) impact or collision is the focus, (3) you need to find an average force over a time interval, or (4) momentum conservation simplifies the problem. Many exam problems intentionally offer 'trap' data (time when only ΣF = ma is needed, or distance when only impulse–momentum applies) to test whether you can filter the given information and select the most efficient path. Practicing this discrimination is as important as executing the calculations correctly.

ΣF = ma is the local (instantaneous) statement of dynamics, while work–energy is its integral over a distance. If you integrate ΣF = ma over displacement ds, you get ∫F·ds = m∫a·ds = m∫(dv/dt)·ds = m∫v dv = (1/2)m(v₂² – v₁²) = ΔKE. Both come from the same fundamental law but answer different questions: ΣF = ma answers 'what is the acceleration now?', while work–energy answers 'what is the final speed after traveling this distance?'

The impulse–momentum principle is the integral of ΣF = ma over time: ∫ΣF dt = ∫m a dt = m∫(dv/dt) dt = m(v₂ – v₁) = Δp. Again, both stem from Newton's second law. ΣF = ma tells you the instantaneous acceleration; impulse–momentum tells you the velocity change over a time interval. They are equivalent but used for different problem contexts.

In a collision, the internal forces between bodies are equal and opposite (Newton's third law) and act for the same time interval. By the impulse–momentum principle, the impulse on body 1 is equal and opposite to that on body 2, so their momenta change in opposite ways, and the total momentum is conserved. D'Alembert's principle reformulates the dynamics of each body as an equilibrium including inertia forces; when you sum these for both bodies, the inertia forces are equal and opposite, again enforcing momentum conservation. This connection shows that momentum conservation is not a separate law but a consequence of Newton's laws and the principle of action–reaction.

When only conservative forces (gravity, springs) do work, the total mechanical energy (KE + PE) is constant. The work–energy principle ΔKE = U_net shows that if U_net = 0 (no net work by conservative forces), then ΔKE = 0. This is equivalent to KE₁ + PE₁ = KE₂ + PE₂. Non-conservative forces (friction, air resistance) dissipate energy, making mechanical energy non-conservative. In exam problems, identifying which forces are conservative is crucial for deciding whether energy is conserved.

The coefficient of restitution e directly measures the fractional restitution of kinetic energy in a collision. For a one-dimensional collision, the kinetic energy after collision is e² times the kinetic energy in the center-of-mass frame before collision. Thus, e = 1 (elastic) means no energy loss; e = 0 (plastic) means maximum energy loss (objects stick together). The relationship between e and energy loss is not linear: KE_after/KE_before = (some function of e, masses, and velocities), but the idea is clear—higher e means more energy retained.

The centripetal (normal) equation ΣF_n = m v²/r shows that the net inward force must equal the mass times the normal acceleration. This governs all circular motion: the tension in a rope, the normal force on a banked curve, the magnetic force on a charged particle, etc. By ΣF = ma, the kinetic energy changes only if tangential forces do work; normal forces (perpendicular to motion) do no work and do not change speed—they only change direction. This is why objects can move at constant speed in a circle yet be accelerating (because acceleration is a vector, and changing direction is acceleration).

In systems with constraints (pulley, rigid rod, incline), the motions of different bodies are related. D'Alembert's principle, by introducing inertia forces, converts each body's dynamics into a statics problem, and the constraints (e.g., the pulley constraint: if one block moves down x, the other moves up x) then become geometric constraints in the pseudo-static equilibrium. This often simplifies solving for accelerations in multi-body systems compared to writing separate ΣF = ma equations for each body and then solving the resulting system of differential equations.

When a car brakes on a horizontal road, the friction force F = μN provides deceleration a = μg. Using kinematics (v² = u² – 2as), the stopping distance is s = u²/(2μg). A civil engineer designing a traffic intersection must ensure that the sight distance is at least equal to the stopping distance at the posted speed limit. For a speed u = 30 m/s and μ = 0.7, the stopping distance is s = 30²/(2×0.7×9.81) ≈ 65 m. This is why speed limits in residential areas are lower: they reduce stopping distance, improving safety. The work–energy principle directly gives U_friction = ΔKE, or μmg·s = (1/2)m v², solving for s.

When an earthquake creates ground acceleration a_ground, a building (mass m) experiences an inertia force F_inertia = m·a_ground. By D'Alembert's principle, this is equivalent to applying a horizontal force F to the building. The building's structural members (columns, beams, shear walls) must resist this force. NSCP 2015 (National Structural Code of the Philippines) accounts for seismic forces using the equivalent lateral force method or response spectrum analysis, both of which rely on kinetics principles. The base shear (total lateral force) is V = C_s·W, where C_s depends on the building's dynamic properties (period, damping) and the ground motion. Understanding kinetics is essential for civil engineers to grasp why earthquakes damage structures and how to design for safety (RA 544, Philippine Building Code).

A crane lifting a load undergoes acceleration and deceleration. During acceleration, the tension in the cable T = m(g + a) exceeds the weight mg; during deceleration, T = m(g – a). A poorly designed acceleration (large a) can cause the cable to fail or the load to swing dangerously. The impulse–momentum principle helps engineers determine the time required to lift a load safely without exceeding the cable's tensile strength. For a 1000 kg load lifted with acceleration a = 0.5 m/s², the tension is T = 1000(9.81 + 0.5) = 10,310 N, compared to the static weight of 9,810 N—a 5% overload. Cables are rated for this dynamic loading (e.g., AISC standards for rigging). Work–energy also applies: the work done by the crane must equal the increase in gravitational potential energy plus the kinetic energy imparted to the load.

When a truck (2000 kg at 15 m/s) strikes a stationary car (1000 kg), the coefficient of restitution e determines the final velocities. Using conservation of momentum and the restitution equation, the final velocities can be calculated. If e = 0 (plastic collision, bodies stick), both move at v = (2000×15)/(2000+1000) = 10 m/s. If e = 0.5 (partially elastic), the relative velocity after is 0.5 times the relative velocity before (15 m/s), giving different final velocities. The impulse experienced by the car is Δp = m_car × Δv, determining the deceleration and potential injuries. Airbags are designed to increase the collision duration (Δt), thereby reducing the average deceleration (F = Δp/Δt) and protecting occupants. This is a life-and-death application of kinetics.

In a centrifuge used in laboratories or in geotechnical testing, a rotating container spins samples at high speed. The normal (centripetal) acceleration a_n = ω²r (where ω is angular velocity and r is radius) can reach many times g. A sample at radius r = 0.1 m rotating at ω = 100 rad/s experiences a_n = 100²×0.1 = 1000 m/s² ≈ 100g. The required centripetal force is F_n = m·a_n. This force comes from the container walls pushing inward on the sample. By Newton's third law, the sample pushes outward on the container with the same force, creating stress in the container material. In geotechnical centrifuge testing (e.g., cone penetration testing under elevated g-levels), this principle is used to model deep-soil behavior at small scale. ΣF_n = m v²/r is the governing equation.

On a banked curve, gravity (mg downward) and the normal force (perpendicular to the road surface) combine to provide the centripetal force needed for circular motion. For a curve of radius r and banking angle θ, if a vehicle travels at the 'design speed' v_design, the banking is such that the required centripetal force is provided by the component of normal force, with minimal reliance on friction. The relationship is tan θ = v_design²/(gr). For a radius of 100 m and design speed of 20 m/s, tan θ = 20²/(100×9.81) ≈ 0.408, giving θ ≈ 22°. At this angle and speed, a frictionless road is safe. However, if vehicles travel faster or slower than the design speed, friction is needed to prevent skidding. Slower vehicles might slide downward if friction is insufficient; faster vehicles might slide outward. This is why posted speed limits on curves are strictly enforced and why curve radii must be adequate for the design speed (DPWH, NSCP standards).

In machinery, friction in bearings, gears, and sliding contacts dissipates energy as heat. The work done by friction is U_f = –μN·d, where d is the sliding distance. Over a long operation period, the total work dissipated is the integral of this. For example, a conveyor belt moving at v = 2 m/s with a friction coefficient μ = 0.3 between the belt and the load (mass m = 100 kg) dissipates energy at a rate P = F·v = μmg·v = 0.3×100×9.81×2 ≈ 588 W ≈ 0.59 kW. Over 8 hours of operation, the energy dissipated is 588 × 8 × 3600 ≈ 17 MJ, requiring cooling systems or contributing to bearing wear. Engineers must account for this in motor sizing and maintenance planning. The work–energy principle is the tool for calculating energy dissipation.

Pedestrian bridges, buildings, and suspension structures oscillate under wind, seismic, or human-induced loads. A simplified model is a pendulum: a mass at the end of a rope or a building sway represented by a spring-mass system. Using energy conservation, if a mass is raised to height h, its potential energy mgh converts to kinetic energy (1/2)m v² at the lowest point, giving v = √(2gh). For a small oscillation of amplitude A, the maximum speed is v_max = ω·A, where ω is the angular frequency. Understanding these dynamics is crucial for assessing bridge safety during construction (e.g., when temporary supports are removed) and for damping design (shock absorbers, tuned mass dampers). The work–energy principle shows that the peak speed is related to the energy input; the impulse–momentum principle helps size dampers to safely dissipate this energy over time (increasing Δt to reduce F).