Dynamics Essentials for the FE Exam
While statics deals with bodies at rest, dynamics deals with bodies in motion. Dynamics problems on the FE exam require you to analyze the motion of particles and rigid bodies under the influence of forces. The two main branches of dynamics are kinematics (describing motion without considering forces) and kinetics (relating forces to motion).
Kinematics of Particles
Rectilinear (Straight-Line) Motion
For motion along a straight line, the key relationships are:
- Velocity: v = ds/dt (rate of change of position)
- Acceleration: a = dv/dt = d2s/dt2 (rate of change of velocity)
- Also useful: a ds = v dv (eliminates time from the equations)
For constant acceleration, the kinematic equations are:
- v = v0 + a*t
- s = s0 + v0*t + (1/2)*a*t^2
- v^2 = v0^2 + 2*a*(s - s0)
These equations are used in projectile motion, falling objects, and vehicle motion problems.
Curvilinear Motion
For motion along a curved path, you can use different coordinate systems:
- Rectangular (x-y): Decompose motion into x and y components. Each component follows straight-line kinematics independently. Used for projectile motion problems.
- Normal-tangential (n-t): The tangential direction is along the path, and the normal direction points toward the center of curvature. Tangential acceleration changes speed; normal acceleration changes direction. a_n = v^2/rho, where rho is the radius of curvature.
- Polar (r-theta): Used when motion is described relative to a fixed point with radial and angular components.
Projectile Motion
A classic kinematics topic. For projectiles launched at angle theta with initial velocity v0:
- Horizontal: x = v0*cos(theta)*t (no horizontal acceleration, ignoring air resistance)
- Vertical: y = v0*sin(theta)*t - (1/2)*g*t^2
- Maximum height: H = v0^2*sin^2(theta) / (2*g)
- Range: R = v0^2*sin(2*theta) / g
Kinetics: Newton's Second Law
Newton's second law (F = ma) is the fundamental equation of kinetics. For a particle:
- Sum of F_x = m * a_x
- Sum of F_y = m * a_y
The approach is to draw a free body diagram showing all forces, then apply Newton's second law in each direction to find unknown forces or accelerations.
Work-Energy Method
The work-energy theorem states: The net work done on a particle equals its change in kinetic energy.
W_net = (1/2)*m*v2^2 - (1/2)*m*v1^2
This method is particularly useful when you need to relate velocity to position without knowing the time. Common applications include:
- Objects sliding down inclines
- Spring-mass systems
- Problems involving friction (where friction does negative work)
When conservative forces are the only forces doing work, conservation of energy applies: KE1 + PE1 = KE2 + PE2.
Impulse-Momentum Method
The impulse-momentum theorem states: The impulse applied to a particle equals its change in momentum.
F * delta_t = m * v2 - m * v1
This method is ideal for problems involving:
- Collisions (elastic and inelastic)
- Impact forces
- Variable forces over time
For collisions, conservation of momentum applies: m1*v1 + m2*v2 = m1*v1' + m2*v2' (primes denote post-collision velocities).
Rigid Body Dynamics
For rigid bodies (as opposed to particles), you must also account for rotation:
- Rotation about a fixed axis: Sum of moments = I * alpha, where I is the mass moment of inertia and alpha is angular acceleration.
- General plane motion: Combine translation of the center of mass with rotation about the center of mass.
- Kinematic relationships: v = omega * r, a_t = alpha * r, a_n = omega^2 * r
Choosing the Right Method
On the FE exam, knowing which method to use is as important as knowing how to use it:
- Use Newton's second law when you need to find acceleration or forces at a specific instant.
- Use work-energy when you know positions and need velocities (or vice versa), and time is not involved.
- Use impulse-momentum when you know forces and time intervals, or when dealing with collisions.
Practice Strategy
Dynamics problems require you to think about which method is most efficient before you start calculating. Practice identifying the correct approach for each problem type, and work through problems using all three methods to understand when each is most effective.
Dynamics is a topic where conceptual understanding and method selection matter as much as calculation skills. Invest time in understanding the physics behind the equations, and the problem-solving will follow.