Mechanical Comprehension
Newton's Laws Applied to Machines
First Law (Inertia): An object at rest stays at rest; a moving object continues at constant velocity — unless a net external force acts on it. More mass = more inertia = harder to start or stop.
Second Law: F = ma. Net force = mass × acceleration. Same force on a heavier object → less acceleration. Measured in Newtons (N).
Third Law: Every action has an equal and opposite reaction. A rocket expels gas backward; the reaction pushes the rocket forward.
Equilibrium: Net force = 0. A stationary bridge is in static equilibrium — gravity pulls down, supports push up, everything balances.
Work and Energy
Work: W = F × d × cos(θ). Only the force component parallel to motion does work. Carrying a weight horizontally across a level floor at constant height does no work — force (upward) is perpendicular to motion (horizontal). Measured in joules (J).
Kinetic energy (KE): KE = ½mv². Doubling speed quadruples KE; tripling speed increases KE by 9×.
Gravitational potential energy (PE): PE = mgh. Depends on mass, gravity, and height above a reference point.
Conservation of energy: In the absence of friction, PE + KE = constant. A ball dropped from height h converts all PE to KE just before impact: mgh = ½mv².
Power: P = W / t = F × v. Rate of doing work. Measured in watts (W). 1 horsepower = 746 W.
Torque
Torque (τ): The rotational equivalent of force. τ = F × d, where d is the perpendicular distance from the pivot (moment arm or lever arm). Measured in Newton-meters (Nm) or foot-pounds (ft-lb).
A longer wrench handle = longer moment arm = more torque with the same force.
Balanced lever: F₁ × d₁ = F₂ × d₂ (effort × effort arm = load × load arm).
Mechanical Advantage
Mechanical advantage (MA): MA = output force / input force. How many times a machine multiplies your force.
All simple machines conserve energy (ideally): what you gain in force you lose in distance — and vice versa.
- MA > 1: force is multiplied (trade: push a longer distance)
- MA = 1: no force gain (just changes direction)
- MA < 1: less force output, but you gain speed or distance
Efficiency: Real machines lose energy to friction. Efficiency (%) = useful work out / work in × 100.
Levers
A rigid bar pivoting on a fulcrum (pivot point). Three classes based on fulcrum position:
| Class | Fulcrum position | Examples | MA |
|---|---|---|---|
| I | Between effort and load | Seesaw, crowbar, scissors | Can be > 1, = 1, or < 1 |
| II | Load between fulcrum and effort | Wheelbarrow, nutcracker, bottle opener | Always > 1 |
| III | Effort between fulcrum and load | Tweezers, fishing rod, human forearm | Always < 1 (gain speed) |
MA = effort arm length / load arm length
Example: Effort arm 3 m, load arm 1 m → MA = 3. Apply 100 N → lifts 300 N.
Pulleys
A wheel with a groove for a rope. Pulleys redirect force and can multiply it.
- Fixed pulley: Attached to ceiling. Changes direction only. MA = 1.
- Movable pulley: Attached to the load. Two rope segments support the load. MA = 2.
- Compound (block and tackle): Multiple pulleys. MA = number of rope segments supporting the load.
- 3 segments: MA = 3 (lift 300 N with 100 N, but must pull 3 m of rope per 1 m load rises)
- 4 segments: MA = 4
Inclined Plane (Ramp)
A tilted flat surface that lets you raise a load over a longer horizontal distance instead of straight up.
MA = length of slope / vertical height
A ramp 5 m long and 1 m high: MA = 5. Push with 1/5 the weight — but over 5× the distance.
More gradual ramp = greater MA. Steeper ramp = less MA.
Wedge, Screw, and Wheel & Axle
Wedge: Two inclined planes back-to-back. Splits, cuts, or secures. A thinner, longer wedge = greater MA. Examples: axe blade, knife, chisel, doorstop.
Screw: An inclined plane wrapped helically around a cylinder. Converts rotational force (torque) into linear motion. Pitch = distance between threads (advance per full rotation). Smaller pitch = more MA but more turns needed. MA = 2πr / pitch.
Wheel and axle: Large wheel + smaller axle rotate together. Turning the large wheel with small force creates large force at the axle. MA = wheel radius / axle radius. Examples: steering wheel, doorknob, winch, screwdriver handle.
Gears
Toothed wheels that mesh to transmit rotational motion and torque.
Key principle: Both gears have the same linear (tangential) speed at the mesh point.
Gear ratio = teeth on driven gear / teeth on driver gear
- Small driver → large driven: driven turns slower, but produces more torque (torque multiplication).
- Large driver → small driven: driven turns faster, but less torque (speed increase).
- Meshing gears always rotate in opposite directions. Add an idler gear to maintain the same direction.
Example: Driver = 40 teeth, driven = 10 teeth. Gear ratio = 10/40 = 1/4. Driven spins 4× faster with 1/4 the torque.
Bevel gears: Conical shape; transmit rotation between perpendicular (non-parallel) shafts (e.g., car differential).
Worm gear: A screw-shaped driver meshing with a helical gear. Very high gear reduction. Self-locking — the gear cannot back-drive the worm.
Pressure and Hydraulics
Pressure: P = F / A (force per unit area). Measured in pascals (Pa) or psi. A concentrated force on a small area = high pressure.
Pascal's Principle: Pressure applied to an enclosed, incompressible fluid is transmitted equally in all directions throughout the fluid.
Hydraulic force multiplication: F₁/A₁ = F₂/A₂
Small piston (2 cm²) with 10 N applied → pressure = 5 N/cm². Large piston (20 cm²) → force = 5 × 20 = 100 N. Force multiplied 10×.
Trade-off: The small piston must move 10 cm for the large piston to move 1 cm. Conservation of energy always applies.
Applications: car jacks, hydraulic brakes, excavators, hydraulic presses.
Friction
Static friction: Prevents an object from starting to move. Usually greater than kinetic friction.
Kinetic (sliding) friction: Resists motion once the object is moving. Always acts opposite to the direction of motion.
Friction force: f = μN, where μ is the coefficient of friction (property of the two surfaces) and N is the normal force (perpendicular to surface).
Rolling friction is much less than sliding friction — this is why wheels were invented.
Lubrication (oil, grease) reduces friction by separating surfaces with a thin film, preventing direct metal-on-metal contact.
Momentum and Collisions
Momentum (p): p = mv (mass × velocity). Measured in kg·m/s.
Conservation of momentum: In a closed system (no external forces), total momentum before = total momentum after any collision.
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
Elastic collision: Kinetic energy conserved; objects bounce off each other (ideal billiard balls).
Perfectly inelastic collision: Objects stick together; kinetic energy is not conserved (some becomes heat/deformation).
Impulse: Change in momentum = F × t. Increasing impact time reduces peak force — why airbags and crumple zones save lives.
Centripetal Force and Circular Motion
Any object moving in a circle constantly changes direction → it is constantly accelerating → requires a net force directed toward the center.
Centripetal force: F_c = mv²/r. Increases with the square of velocity. Decreases with a larger radius.
Provided by: string tension (ball on string), friction (car turning), gravity (satellite orbit), normal force (banked curve).
Centrifugal "force": The apparent outward feeling in a turning car. This is not a real force — it is the result of inertia resisting the change in direction. In an inertial (non-rotating) reference frame, you simply tend to continue straight while the car turns.
Material Properties
- Strength: Resistance to deformation. Tensile = being pulled apart. Compressive = being crushed.
- Elasticity: Returns to original shape after deformation (rubber, steel springs).
- Plastic deformation: Permanent deformation that does not return.
- Brittle: Breaks without significant deformation (glass, cast iron).
- Ductile: Can be drawn into wire without breaking (copper, gold).
- Malleable: Can be hammered flat without cracking (gold, aluminum).
- Hardness: Resistance to scratching or indentation.
- Density: Mass per unit volume (kg/m³).
Structural forces: - Tension: Pulling/stretching force (suspension bridge cables, ropes). - Compression: Pushing/squeezing force (columns, arches). - Shear: Layers sliding past each other (scissors cutting, bolt being sheared off).
Springs and Elastic Energy
Hooke's Law: F = kx, where k is the spring constant (stiffness, in N/m) and x is compression or extension from rest (in meters). A stiffer spring has a larger k.
Elastic potential energy stored in a spring: PE = ½kx²
Example: Spring with k = 200 N/m compressed 0.1 m: PE = ½ × 200 × 0.01 = 1 J.
Doubling the compression quadruples the stored energy (x is squared).