Dynamics on the FE Exam
Dynamics is the study of bodies in motion and the forces that cause that motion. It appears on the FE Mechanical exam (7-11%), FE Other Disciplines exam (7-11%), and FE Civil exam (3-5%). The subject divides into two main areas: kinematics (the geometry of motion) and kinetics (the relationship between forces and motion).
Particle Kinematics
Rectilinear Motion
For motion along a straight line, position, velocity, and acceleration are related through derivatives and integrals. Velocity is the time derivative of position, and acceleration is the time derivative of velocity. For constant acceleration, use the three standard kinematic equations relating initial velocity, final velocity, acceleration, displacement, and time.
Curvilinear Motion
For motion along a curved path, use rectangular (x-y), normal-tangential (n-t), or polar (r-theta) coordinate systems depending on the problem geometry. In the normal-tangential system, the tangential acceleration changes the speed while the normal acceleration changes the direction. Normal acceleration equals the velocity squared divided by the radius of curvature.
Projectile Motion
Projectile motion is a special case of curvilinear motion with constant gravitational acceleration in the vertical direction and zero acceleration in the horizontal direction. Know how to calculate range, maximum height, and time of flight from initial velocity and launch angle.
Newton's Second Law
Newton's Second Law (F = ma) is the foundation of kinetics. For a particle, the sum of all forces equals the mass times the acceleration. Apply this principle by drawing a free body diagram and a kinetic diagram (showing the ma vector), then writing equations of motion in each coordinate direction.
For systems of particles connected by ropes, pulleys, or rigid links, write the equations of motion for each particle separately and add constraint equations relating their accelerations. The number of equations must equal the number of unknowns.
Work and Energy
The work-energy theorem states that the net work done on a particle equals its change in kinetic energy. This method is powerful when you need to relate forces and displacements without knowing the time involved.
For conservative systems, use conservation of energy: the sum of kinetic and potential energy is constant. Potential energy includes gravitational potential energy (mgh) and elastic potential energy (0.5 times k times x squared for springs). The work-energy approach is especially useful for problems involving gravity, springs, and varying forces.
Impulse and Momentum
The impulse-momentum theorem states that the impulse (force times time) equals the change in linear momentum (mass times velocity). This method is ideal for problems involving impacts, collisions, and time-dependent forces.
Conservation of linear momentum applies when the net external force on a system is zero. For collisions, use conservation of momentum along with the coefficient of restitution (which relates the relative velocities before and after impact). Know the difference between perfectly elastic collisions (coefficient of restitution equals 1, kinetic energy is conserved), perfectly plastic collisions (coefficient equals 0, objects stick together), and partially elastic collisions.
Rigid Body Dynamics
For rigid bodies, you must account for both translational and rotational motion. The sum of forces equals the mass times the acceleration of the center of mass, and the sum of moments about the center of mass equals the mass moment of inertia times the angular acceleration.
Mass moments of inertia differ from area moments of inertia. Know the mass moment of inertia for common shapes (slender rod, disk, sphere, rectangular plate) and how to use the parallel axis theorem for mass moments of inertia.
Vibrations
Free vibration of a single-degree-of-freedom system (mass-spring or mass-spring-damper) produces oscillatory motion characterized by natural frequency, period, and amplitude. The natural frequency of an undamped system equals the square root of the spring constant divided by the mass.
For damped systems, the damping ratio determines whether the response is underdamped (oscillatory), critically damped (fastest return to equilibrium without oscillation), or overdamped (slow return without oscillation). Know how to calculate the damped natural frequency and the logarithmic decrement for underdamped systems.
Exam Tips
Choose the right analysis method for each problem. Use Newton's Second Law when forces and accelerations are the primary unknowns. Use work-energy when forces and displacements are known. Use impulse-momentum for impact and collision problems. Selecting the correct method can reduce a complex problem to a simple calculation.