Forces and Energy

Forces as Vectors

Force has magnitude and direction. That makes it a vector. You can't just add forces like numbers—you have to account for direction. 10 N east plus 10 N west doesn't equal 20 N; it equals zero (forces cancel). 10 N east plus 10 N north equals √(10² + 10²) ≈ 14.1 N northeast (vector addition, Pythagorean theorem).

Structural analysis, FEA, machine design—all rely on decomposing forces into components, analyzing each direction independently, then recombining. Mess this up and your load path analysis is wrong, which means your design is wrong. Vector handling isn't optional.

Newton's Laws

First law: If net force is zero, velocity doesn't change. Object at rest stays at rest. Object moving at constant velocity keeps moving at constant velocity. ΣF = 0 → a = 0. This is equilibrium—the foundation of statics.

Second law: If net force isn't zero, the object accelerates. ΣF = ma. Force causes acceleration, which is the rate of change of velocity. More force → more acceleration (for fixed mass). More mass → less acceleration (for fixed force).

First law covers statics (structures, beams, trusses). Second law covers dynamics (vibration, robotics, vehicles, anything that moves). Everything in mechanical engineering starts with one of these two.

Free-Body Diagrams

A free-body diagram (FBD) shows every force acting on an object. You isolate the object from its surroundings, then draw arrows representing each force—gravity, normal force, tension, friction, applied forces, everything. If it's not on the diagram, it doesn't go in the equations.

Most force analysis errors come from incomplete FBDs. Forgot to draw friction? Your equations won't include it, and your answer will be wrong. Drew a force that doesn't actually exist? Your answer will be wrong in a different way. FBDs aren't busywork—they're how you translate a physical situation into mathematical form without missing anything.

The rule: if a force acts on the object, it appears on the FBD. If it doesn't act on the object (like forces the object exerts on other things), it doesn't appear. Draw it correctly and the rest is just algebra. Draw it wrong and no amount of math will save you.

Work and Kinetic Energy

Work is what happens when a force moves something. W = F · d (force dot displacement). Only the component of force in the direction of motion counts. Push perpendicular to motion? No work done.

Kinetic energy (KE = ½mv²) is energy of motion. Work changes kinetic energy—that's the work-energy theorem. Push something over a distance and it speeds up. The work you did equals the change in kinetic energy. What makes this powerful: you skip the intermediate steps. Don't need velocity at every instant, just initial and final. Don't need time. Just work in, kinetic energy out.

Potential Energy

Potential energy is stored energy. Position-dependent. Lift a mass against gravity? PE = mgh gets stored. Compress a spring? PE = ½kx² gets stored. Release it and potential converts to kinetic as the object moves.

This is the appeal of energy methods. They often bypass force analysis entirely. When you care about initial and final states but not the path in between, energy is usually faster.

Conservation of Energy

Ignore friction and other losses. Total mechanical energy stays constant. KE₁ + PE₁ = KE₂ + PE₂. Energy shifts between forms but the sum doesn't change.

No differential equations needed. Just balance energy. This is how you analyze vibration, how Lagrangian mechanics works, how you model mechanisms without tracking every force. Energy conservation isn't a shortcut—it's often the cleanest approach. Use it whenever friction and losses are negligible or accounted for separately.

Worked Example: Force vs Energy Approaches

Takeaway: Both approaches work. Force analysis tells you everything at every instant. Energy analysis tells you initial vs final states with less calculation. Choose based on what the problem gives you and what it asks for.