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10.5 Forces, Friction, Work, Power, Energy Conservation, and Wave Mechanics
CAT-ASVAB Computer Adaptive ASVAB
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1 Chapter 1: CAT-ASVAB Overview, Adaptive Mechanics & Test Strategy
1.1 CAT-ASVAB Architecture, MEPS Administration & Subtest Structure 1.2 Computer Adaptive Testing (CAT), IRT Engine & Pacing Rules 1.3 AFQT Scoring Formula, VE Weighting & Service Branch Minimums 1.4 Military Composite Line Scores & MOS/AFSC/Rating Classification
2 Chapter 2: Word Knowledge (WK)
2.1 Greek and Latin Roots, Prefixes, and Suffixes 2.2 Context Clues, Secondary Meanings, and Nuanced Synonyms 2.3 Military, Technical, and Formal Vocabulary in Context 2.4 High-Frequency Word Bank and the 36-Second Word-Attack Protocol
3 Chapter 3: Paragraph Comprehension (PC)
3.1 Identifying Main Ideas, Central Themes, and Author Purpose 3.2 Supporting Details, Factual Extraction, and Structural Analysis 3.3 Inferences, Contextual Meaning, and Tone Evaluation
4 Chapter 4: Arithmetic Reasoning (AR)
4.1 Translating Complex Word Problems into Algebraic Expressions 4.2 Ratios, Proportions, Rates, and Combined Work Problems 4.3 Percentages, Interest, Profit, and Financial Math 4.4 Applied Geometric Word Problems (Area, Perimeter, Volume)
5 Chapter 5: Mathematics Knowledge (MK)
5.1 Number Properties, Exponents, Radicals, and Polynomials 5.2 Linear Equations, Quadratic Equations, and Inequalities 5.3 Geometric Principles, Coordinate Geometry, and Trigonometry Fundamentals 5.4 Basic Probability, Permutations, Combinations, and Statistics 5.5 Fractions, Rational Expressions, and Non-Calculator Computation
6 Chapter 6: General Science (GS)
6.1 Cellular Biology, Human Anatomy, and Biological Systems 6.2 Chemistry Fundamentals, Atomic Structure, and Reactions 6.3 Classical Physics, Energy, Forces, and Motion 6.4 Earth Science, Meteorology, Astronomy, and Environmental Science 6.5 Scientific Method, Measurement Systems, and Laboratory Instruments
7 Chapter 7: Electronics Information (EI)
7.1 Voltage, Current, Resistance, Ohm's Law, and Power 7.2 Circuit Analysis: Series, Parallel, and Combination Circuits with Kirchhoff's Laws 7.3 Electronic Components, Semiconductor Devices, and Schematic Symbols 7.4 Magnetism, Electromagnetism, Motors, and Generators 7.5 Circuit Protection, Wiring, Measurement Instruments, and Electrical Safety
8 Chapter 8: Auto Information (AI)
8.1 Internal Combustion Engines and the Four-Stroke Cycle 8.2 Powertrain, Transmission, Drivetrain, and Fuel Systems 8.3 Braking Systems, Suspension, Steering, and Electrical Diagnostics
9 Chapter 9: Shop Information (SI)
9.1 Hand Tools, Measuring Instruments, and Cutting Implements 9.2 Power Tools, Fasteners, Hardware, and Joining Techniques 9.3 Woodworking, Metalworking, Materials, and Workshop Safety
10 Chapter 10: Mechanical Comprehension (MC)
10.1 Simple Machines: Levers, Classes, and Mechanical Advantage 10.2 Pulleys, Wheel and Axle Systems, and Inclined Planes 10.3 Gears, Belt and Chain Drives, Cams, and Torque Transmission 10.4 Fluid Power, Hydraulics, Pneumatics, and Buoyancy (Pascal & Archimedes) 10.5 Forces, Friction, Work, Power, Energy Conservation, and Wave Mechanics
11 Chapter 11: Assembling Objects (AO)
11.1 Spatial Relationship Fundamentals & 2D Silhouette Synthesis 11.2 Connection Point Problems, Interior Seams, and Labeled Points 11.3 High-Speed Pacing, Elimination Tactics, and Visual Pattern Recognition
10.5 Forces, Friction, Work, Power, Energy Conservation, and Wave Mechanics
Key Takeaways Explain This Topic AI
Newton's Three Laws of Motion establish classical dynamics: Inertia (F_net = 0 implies a = 0), Force-Acceleration (F_net = m · a), and Action-Reaction (F_A to B = -F_B to A acting on distinct bodies with equal magnitude in opposite directions).
Frictional resistance opposes relative surface motion (F_f = μ · N), where the coefficient of static friction (μ_s) is always greater than the coefficient of kinetic friction (μ_k), and normal force on an incline is reduced by N = m g cos(θ).
Mechanical Work (W = F · d · cos θ) requires parallel force and displacement; forces applied perpendicular to motion (θ = 90°) perform zero work. Mechanical Power (P = W / t = F · v) measures work rate (1 HP = 550 ft-lb/s = 746 W).
In conservative mechanical systems, total energy is conserved (E_total = PE + KE = constant); gravitational potential energy (PE = mgh) converts completely into kinetic energy (KE = 1/2 m v²), yielding terminal free-fall velocity v = √(2gh) independent of mass.
Wave mechanics follow the universal wave speed equation (v = f · λ); transverse waves oscillate perpendicular to energy propagation (light, electromagnetic waves, surface water waves), while longitudinal waves oscillate parallel via alternating compressions and rarefactions (sound waves, seismic P-waves).
Last updated: August 2026
10.5 Forces, Friction, Work, Power, Energy Conservation, and Wave Mechanics
Core Principle: Physical mechanics governs how forces interact, how friction resists motion, how potential and kinetic energy transition within mechanical systems, and how energy propagates through media as oscillatory waves. On the CAT-ASVAB Mechanical Comprehension subtest, physics questions evaluate your mastery of Newton's Three Laws of Motion, static vs. kinetic friction thresholds, work and horsepower calculations, gravitational potential and kinetic energy conversions, and wave mechanics formulas ( v = f ⋅ λ v = f \cdot \lambda v = f ⋅ λ ).
Newton's Three Laws of Motion
Classical Newtonian dynamics provides the structural foundation for all mechanical engineering and military hardware systems. +-----------------------------------------------------------------------------------------+
- | NEWTON'S THREE LAWS OF MOTION |
- +-------------------+----------------------------+----------------------------------------+
- | Law of Motion | Formal Physical Principle | Real-World & Military Applications |
- +-------------------+----------------------------+----------------------------------------+
- | Newton's 1st Law | Law of Inertia: An object | Vehicle seatbelts, cargo tie-down |
- | (Inertia) | remains at rest or in uni- | straps, centrifugal oil filters, |
- | | form straight-line motion | projectile trajectory upon barrel exit |
- | | unless acted on by net F. | |
- +-------------------+----------------------------+----------------------------------------+
- | Newton's 2nd Law | Fundamental Dynamics Law: | Calculating vehicle braking force, |
- | (F = m · a) | Acceleration is directly | artillery shell propulsion, aircraft |
- | | proportional to net force | catapult launch acceleration |
- | | and inversely to mass. | |
- +-------------------+----------------------------+----------------------------------------+
- | Newton's 3rd Law | Action-Reaction: For every | Rocket engine thrust, firearm recoil, |
- | (Action/Reaction) | action force, there is an | helicopter tail rotor counter-torque, |
- | | equal and opposite reaction| naval propeller driving against water |
- | | on a DIFFERENT body. | |
- +-------------------+----------------------------+----------------------------------------+
- 1. The First Law: Inertia & Net Force Equilibrium
Inertia is the inherent resistance of any physical mass to a change in its state of motion. Mass is the direct quantitative measure of inertia.
Static Equilibrium: When all forces acting on a body balance out to zero ( Σ F x = 0 , Σ F y = 0 \Sigma F_x = 0, \Sigma F_y = 0 Σ F x = 0 , Σ F y = 0 ), the object experiences zero net acceleration ( a = 0 a = 0 a = 0 ). An object in equilibrium is either completely stationary or moving at a constant velocity in a straight line.
2. The Second Law: Force, Mass & Acceleration ( F = m ⋅ a F = m \cdot a F = m ⋅ a )
Formula: Σ F = m ⋅ a ⟹ a = Σ F m ⟹ m = Σ F a \Sigma F = m \cdot a \implies a = \frac{\Sigma F}{m} \implies m = \frac{\Sigma F}{a} Σ F = m ⋅ a ⟹ a = m Σ F ⟹ m = a Σ F
Units:
SI Metric: Mass in kilograms ( kg \text{kg} kg ), Acceleration in m/s 2 \text{m/s}^2 m/s 2 , Force in Newtons ( N = kg ⋅ m/s 2 \text{N} = \text{kg}\cdot\text{m/s}^2 N = kg ⋅ m/s 2 ).
US Customary: Weight ( W = m ⋅ g W = m \cdot g W = m ⋅ g ) in pounds ( lb \text{lb} lb ), Acceleration of gravity g = 32.2 ft/s 2 g = 32.2\text{ ft/s}^2 g = 32.2 ft/s 2 . Mass is measured in slugs ( slug = lb / 32.2 ft/s 2 \text{slug} = \text{lb} / 32.2\text{ ft/s}^2 slug = lb /32.2 ft/s 2 ).
Calculation Example: A 3 , 220 -lb 3,220\text{-lb} 3 , 220 -lb military tactical vehicle accelerates from rest at 4 ft/s 2 4\text{ ft/s}^2 4 ft/s 2 . Find the net forward driving force required: m = W g = 3 , 220 lbs 32.2 ft/s 2 = 100 slugs m = \frac{W}{g} = \frac{3,220\text{ lbs}}{32.2\text{ ft/s}^2} = 100\text{ slugs} m = g W = 32.2 ft/s 2 3 , 220 lbs = 100 slugs F = m ⋅ a = 100 slugs × 4 ft/s 2 = 400 lbs F = m \cdot a = 100\text{ slugs} \times 4\text{ ft/s}^2 = 400\text{ lbs} F = m ⋅ a = 100 slugs × 4 ft/s 2 = 400 lbs
3. The Third Law: Action and Reaction
Every force is an interaction between two distinct bodies. If Body A exerts force F A → B F_{A \to B} F A → B on Body B, Body B simultaneously exerts an equal and opposite force F B → A F_{B \to A} F B → A on Body A: F A → B = − F B → A F_{A \to B} = -F_{B \to A} F A → B = − F B → A
ASVAB Exam Trap: Action and reaction forces never cancel each other out because they act on two separate, distinct objects . When a rifle fires a bullet, the forward force acts on the bullet (accelerating it to high speed due to its small mass), while the equal backward recoil force acts on the rifle (pushing back against the shooter's shoulder).
Friction Dynamics: Normal Force, Static vs. Kinetic Friction
Friction is the resistive contact force that opposes the relative sliding or rolling motion between two surfaces in physical contact. +-----------------------------------------------------------------------------------------+
- | STATIC VS. KINETIC FRICTION |
- +--------------------------+------------------------------+-------------------------------+
- | Characteristic | Static Friction (F_s) | Kinetic (Sliding) Friction(F_k|
- +--------------------------+------------------------------+-------------------------------+
- | State of Motion | Surfaces are stationary | Surfaces are actively sliding |
- | | relative to each other | past one another |
- +--------------------------+------------------------------+-------------------------------+
- | Governing Formula | F_s,max = μ_s · N | F_k = μ_k · N |
- +--------------------------+------------------------------+-------------------------------+
- | Relative Magnitude | ALWAYS HIGHER (μ_s > μ_k) | ALWAYS LOWER (μ_k < μ_s) |
- +--------------------------+------------------------------+-------------------------------+
- | Physical Cause | Interlocking surface asper- | Shearing of surface bonds |
- | | ities and microscopic bonds | while moving over peaks |
- +--------------------------+------------------------------+-------------------------------+
- | Real-World Implication | Takes more force to START | Takes less force to KEEP |
- | | moving a heavy crate | an object sliding once moving |
+--------------------------+------------------------------+-------------------------------+ APPLIED FORCE VS. FRICTIONAL RESISTANCE CURVE
- Friction Force (F_f)
- ▲
- │ Peak Static Friction: F_s,max = μ_s · N
- │ /│
- │ / │
- │ / │ Kinetic Friction: F_k = μ_k · N (Constant)
- │ / └───────────────────────────────
- │ Static Region / Sliding (Kinetic) Region
- │ (F_s = F_app)/
- └──────────────/──────────────────────────────────────► Applied Force (F_app)
- ◄────── Static ──────►◄──── Kinetic ───►
- The Normal Force ( N N N )
The Normal Force ( N N N ) is the perpendicular compressive contact force exerted by a supporting surface on an object:
Horizontal Surface: N = W = m ⋅ g N = W = m \cdot g N = W = m ⋅ g
Inclined Plane at Angle θ \theta θ : N = m ⋅ g ⋅ cos ( θ ) N = m \cdot g \cdot \cos(\theta) N = m ⋅ g ⋅ cos ( θ ) Parallel Downhill Gravity Force: F ∥ = m ⋅ g ⋅ sin ( θ ) \text{Parallel Downhill Gravity Force: } F_{\parallel} = m \cdot g \cdot \sin(\theta) Parallel Downhill Gravity Force: F ∥ = m ⋅ g ⋅ sin ( θ ) Frictional Resistance on Incline: F f = μ ⋅ N = μ ⋅ m ⋅ g ⋅ cos ( θ ) \text{Frictional Resistance on Incline: } F_f = \mu \cdot N = \mu \cdot m \cdot g \cdot \cos(\theta) Frictional Resistance on Incline: F f = μ ⋅ N = μ ⋅ m ⋅ g ⋅ cos ( θ )
Rolling Friction vs. Sliding Friction
Rolling friction occurs when a rounded wheel, ball bearing, or roller rolls across a surface. Rolling friction coefficients ( μ r ≈ 0.001 − 0.01 \mu_r \approx 0.001 - 0.01 μ r ≈ 0.001 − 0.01 ) are 10 to 100 times smaller than sliding friction coefficients ( μ k ≈ 0.2 − 0.8 \mu_k \approx 0.2 - 0.8 μ k ≈ 0.2 − 0.8 ). This explains why ball bearings and roller bearings are universally used in vehicle axles, gearboxes, and aircraft turbines.
Work, Energy Conservation & Free-Fall Dynamics
Energy is the capacity to perform mechanical work. In a closed conservative system where non-conservative dissipative forces (such as friction and air drag) are neglected, total mechanical energy ( E t o t a l E_{total} E t o t a l ) remains constant . +-----------------------------------------------------------------------------------------+
- | POTENTIAL VS. KINETIC ENERGY COMPARISON |
- +--------------------------+------------------------------+-------------------------------+
- | Energy Form | Gravitational Potential (PE) | Kinetic Energy (KE) |
- +--------------------------+------------------------------+-------------------------------+
- | Defining State | Energy stored by position | Energy possessed by motion |
- | | or elevation in gravity field| and velocity of mass |
- +--------------------------+------------------------------+-------------------------------+
- | Governing Formula | PE = m · g · h = W · h | KE = 1/2 · m · v² |
- +--------------------------+------------------------------+-------------------------------+
- | Variable Dependency | Linear with height (h) | QUADRATIC with velocity (v²) |
- +--------------------------+------------------------------+-------------------------------+
- | Velocity Doubling Impact | No change | QUADRUPLES (4×) kinetic energy|
- +--------------------------+------------------------------+-------------------------------+
- | Maximum Location | Peak elevation (top of drop) | Maximum velocity (bottom) |
- +--------------------------+------------------------------+-------------------------------+
Conservation of Mechanical Energy: P E i n i t i a l + K E i n i t i a l = P E f i n a l + K E f i n a l PE_{initial} + KE_{initial} = PE_{final} + KE_{final} P E ini t ia l + K E ini t ia l = P E f ina l + K E f ina l
When an object falls from a height h h h from rest ( v i n i t i a l = 0 , K E i n i t i a l = 0 v_{initial} = 0, KE_{initial} = 0 v ini t ia l = 0 , K E ini t ia l = 0 ), all of its potential energy converts into kinetic energy at the bottom ( P E f i n a l = 0 PE_{final} = 0 P E f ina l = 0 ): m ⋅ g ⋅ h = 1 2 m v 2 m \cdot g \cdot h = \frac{1}{2} m v^2 m ⋅ g ⋅ h = 2 1 m v 2
Dividing both sides by mass m m m : g ⋅ h = 1 2 v 2 ⟹ v 2 = 2 g h ⟹ v = 2 g h g \cdot h = \frac{1}{2} v^2 \implies v^2 = 2 g h \implies v = \sqrt{2 g h} g ⋅ h = 2 1 v 2 ⟹ v 2 = 2 g h ⟹ v = 2 g h
Key Exam Takeaway: The impact velocity of a falling object depends ONLY on the drop height ( h h h ) and gravity ( g g g ) —it is completely independent of the object's mass! A 10-lb bowling ball and a 100-lb anvil dropped from the same height hit the ground at the exact same velocity (neglecting air resistance).
The Quadratic Velocity Rule: Braking Distance & Kinetic Energy
Because Kinetic Energy is proportional to the square of velocity ( v 2 v^2 v 2 ) :
If a vehicle doubles its speed from 30 mph 30\text{ mph} 30 mph to 60 mph 60\text{ mph} 60 mph ( 2 × 2\times 2 × ), its kinetic energy increases by 4 × 4\times 4 × ( 2 2 2^2 2 2 ) .
Because braking work is W b r a k e = F f r i c t i o n ⋅ d s t o p = K E W_{brake} = F_{friction} \cdot d_{stop} = KE W b r ak e = F f r i c t i o n ⋅ d s t o p = K E , the minimum required braking distance quadruples ( 4 × 4\times 4 × ) when speed doubles!
Wave Mechanics: Transverse vs. Longitudinal Waves
A wave is a rhythmic disturbance that propagates energy through space or a material medium without transporting physical matter. +-----------------------------------------------------------------------------------------+
- | CORE WAVE ANATOMY & UNITS |
- +-------------------+--------+--------------------+---------------------------------------+
- | Wave Property | Symbol | SI Unit | Definition & Description |
- +-------------------+--------+--------------------+---------------------------------------+
- | Crest | — | — | The highest peak point of the wave |
- | Trough | — | — | The lowest valley point of the wave |
- | Amplitude | A | Meter (m) / Feet | Height from equilibrium centerline to |
- | | | | peak (measures wave energy / volume) |
- | Wavelength | λ | Meter (m) / Feet | Distance between successive identical |
- | (Lambda) | | | points (crest-to-crest or trough) |
- | Frequency | f | Hertz (Hz = 1/s) | Number of complete wave cycles passing|
- | | | | a fixed point per second |
- | Period | T | Second (s) | Time required for 1 full cycle (T=1/f)|
- | Wave Speed | v | m/s or ft/s | Velocity of energy propagation |
- +-------------------+--------+--------------------+---------------------------------------+ Wavelength (λ)
- ◄──────────────────────►
- ▲ ╭───╮ ╭───╮
- Amplitude │ ╱ ╲ ╱ ╲
- (A) ▼ │ Crest │ │ Crest │
- ───────────────┼─────────┼────────┼─────────┼─────────────── Equilibrium Line
- ╲ ╱ Trough ╲ ╱
- ╰─────╯ ╰─────╯
- The Universal Wave Equation
The velocity ( v v v ) of any wave is the product of its frequency ( f f f ) and its wavelength ( λ \lambda λ ): v = f ⋅ λ ⟹ f = v λ ⟹ λ = v f v = f \cdot \lambda \implies f = \frac{v}{\lambda} \implies \lambda = \frac{v}{f} v = f ⋅ λ ⟹ f = λ v ⟹ λ = f v
Transverse Waves vs. Longitudinal Waves +-----------------------------------------------------------------------------------------+
- | TRANSVERSE VS. LONGITUDINAL WAVES |
- +--------------------------+------------------------------+-------------------------------+
- | Wave Characteristic | Transverse Waves | Longitudinal Waves |
- +--------------------------+------------------------------+-------------------------------+
- | Particle Oscillation | PERPENDICULAR (90°) to the | PARALLEL to the direction of |
- | Direction | direction of wave propagation| wave propagation |
- +--------------------------+------------------------------+-------------------------------+
- | Wave Structure | Alternating Crests & Troughs | Alternating Compressions |
- | | | and Rarefactions |
- +--------------------------+------------------------------+-------------------------------+
- | Propagation Media | Vacuum (for EM waves), | Solids, Liquids, and Gases |
- | | Solids, Surface of liquids | (CANNOT propagate in vacuum) |
- +--------------------------+------------------------------+-------------------------------+
- | Real-World Examples | Light waves, radio waves, | Sound waves, ultrasonic sonar,|
- | | radar, microwaves, X-rays, | acoustic vibrations, seismic |
- | | ocean ripples, guitar string | P-waves, compression springs |
+--------------------------+------------------------------+-------------------------------+ Transverse Wave: Particle Motion: ↕ (Up/Down) Wave Propagation: ──► (Right)
- ╭───╮ ╭───╮ ╭───╮
- ╱ ╲ ╱ ╲ ╱ ╲
- │ │ │ │ │ │
- ╰─╯ ╰─╯ ╰─╯
- Longitudinal Wave: Particle Motion: ◄► (Parallel) Wave Propagation: ──► (Right)
- |||||||| | | | |||||||| | | | ||||||||
- Compression Rarefact Compression Rarefact Compression
- Speed of Sound Across Media
Sound is a mechanical longitudinal wave that relies on molecular collisions to transmit energy:
Solids (Steel): Speed ≈ 5 , 000 m/s \approx 5,000\text{ m/s} ≈ 5 , 000 m/s ( 16 , 400 ft/s 16,400\text{ ft/s} 16 , 400 ft/s ) → \rightarrow → Fastest due to high elasticity and tightly bonded molecules.
Liquids (Water): Speed ≈ 1 , 500 m/s \approx 1,500\text{ m/s} ≈ 1 , 500 m/s ( 4 , 900 ft/s 4,900\text{ ft/s} 4 , 900 ft/s ) → \rightarrow → Intermediate speed (used in naval sonar).
Gases (Air at 20 ∘ C 20^\circ\text{C} 2 0 ∘ C ): Speed ≈ 343 m/s \approx 343\text{ m/s} ≈ 343 m/s ( 1 , 125 ft/s 1,125\text{ ft/s} 1 , 125 ft/s ) → \rightarrow → Slowest due to widely spaced gas molecules.
Vacuum (Outer Space): Speed = 0 m/s = 0\text{ m/s} = 0 m/s → \rightarrow → Sound cannot travel in a vacuum because there are no particles to compress.
- Step-by-Step Problem Derivations
- Problem 1: Free-Fall Drop Velocity
A 50-lb artillery shell falls from a gantry hoist platform located 64 feet above the ground. Using g = 32 ft/s 2 g = 32\text{ ft/s}^2 g = 32 ft/s 2 and neglecting air resistance:
What is the shell's potential energy at the top?
What is its velocity just before striking the ground? Solution Steps:
Step 1: PE = W × h = 50 lbs × 64 ft = 3,200 ft-lb.
Step 2: By energy conservation: v = √(2 · g · h) = √(2 · 32 ft/s² · 64 ft) = √(4,096) = 64 ft/s.
Problem 2: Wave Velocity & Sonar Frequency
A naval active sonar system emits an acoustic wave with a frequency of 3 , 000 Hz 3,000\text{ Hz} 3 , 000 Hz through ocean water (speed of sound in seawater = 1 , 500 m/s = 1,500\text{ m/s} = 1 , 500 m/s ).
What is the wavelength ( λ \lambda λ ) of this sonar pulse?
What is the period ( T T T ) of the wave? Solution Steps:
Step 1: λ = v / f = 1,500 m/s / 3,000 Hz = 0.50 meters (50 cm).
Step 2: T = 1 / f = 1 / 3,000 Hz ≈ 0.000333 seconds (0.333 ms).