Slug (sl) - Unit Information & Conversion

Quick Answer

1 slug = 14.5939 kg = 32.174 pounds-mass (lbm)

Defined by Newton's second law: 1 slug is the mass that accelerates at 1 ft/s² when a force of 1 lbf is applied

Formula: 1 lbf = 1 slug × 1 ft/s² (making F = ma work cleanly in imperial units)

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Quick Comparison Table

Mass Unit	Equals 1 Slug
Kilograms	14.5939 kg
Grams	14,593.9 g
Pounds-mass	32.174 lbm
Ounces	514.78 oz
Metric tons	0.0145939 t
Short tons (US)	0.0160870 ton

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Definition

What Is a Slug?

The slug (symbol: sl or slug) is a unit of mass in the Foot-Pound-Second (FPS) system of imperial units. It is defined through Newton's second law of motion (F = ma):

1 slug = 1 lbf / (1 ft/s²)

In words: one slug is the mass that accelerates at one foot per second squared when a force of one pound-force is applied to it.

Exact Value

1 slug = 32.17404855... pounds-mass (lbm) ≈ 32.174 lbm

1 slug = 14.593902937206... kilograms ≈ 14.5939 kg

These values derive from the standard acceleration due to gravity: g = 32.174 ft/s² = 9.80665 m/s².

The Pound Confusion

The imperial system has a fundamental ambiguity: the word "pound" means two different things:

Pound-mass (lbm):

• A unit of mass (quantity of matter)
• An object has the same pound-mass everywhere in the universe
• Symbol: lbm

Pound-force (lbf):

• A unit of force (weight)
• The force exerted by one pound-mass under standard Earth gravity
• Symbol: lbf
• 1 lbf = 1 lbm × 32.174 ft/s² (weight = mass × gravity)

This creates confusion because in everyday language, "pound" can mean either, depending on context. The slug eliminates this ambiguity by serving as an unambiguous mass unit compatible with pound-force.

Why the Slug Matters: Making F = ma Work

Newton's second law: F = ma (Force = mass × acceleration)

Problem with pounds-mass and pounds-force: If you use lbm for mass and lbf for force, Newton's law becomes: F = ma / g_c

where g_c = 32.174 lbm·ft/(lbf·s²) is a dimensional conversion constant—ugly and error-prone!

Solution with slugs: Using slugs for mass and lbf for force, Newton's law works cleanly: F = ma (no extra constants needed!)

Example:

• Force: 10 lbf
• Acceleration: 5 ft/s²
• Mass: F/a = 10 lbf / 5 ft/s² = 2 slugs
• (Or in lbm: mass = 2 slugs × 32.174 = 64.348 lbm)

FPS System

The slug is part of the Foot-Pound-Second (FPS) system, also called the British Gravitational System or English Engineering System:

• Length: foot (ft)
• Force: pound-force (lbf)
• Time: second (s)
• Mass: slug (sl)
• Acceleration: feet per second squared (ft/s²)

This contrasts with the SI system (meter, kilogram, second, newton) and the pound-mass system (foot, pound-mass, second, poundal).

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History

The Imperial Weight-Mass Problem (Pre-1900)

Before the slug was invented, the imperial system created confusion between weight (force due to gravity) and mass (quantity of matter):

Common usage: "Pound" meant weight (what a scale measures on Earth)

• "This weighs 10 pounds" meant 10 pounds-force (10 lbf)

Scientific usage: "Pound" could mean mass (quantity of matter)

• "This has 10 pounds of mass" meant 10 pounds-mass (10 lbm)

The problem: Newton's laws of motion require distinguishing force from mass. Using "pound" for both led to:

• Confusion in physics calculations
• Need for awkward gravitational conversion constants
• Errors in engineering (mixing lbf and lbm)

Arthur Mason Worthington (1852-1916)

Arthur Mason Worthington was a British physicist and professor at the Royal Naval College, Greenwich, known for his pioneering work in:

• High-speed photography of liquid drops and splashes
• Physics education and textbook writing
• Developing clearer terminology for imperial units

Around 1900, Worthington recognized that the imperial system needed a mass unit analogous to the kilogram—a unit that would make Newton's second law (F = ma) work without conversion factors.

The Slug's Introduction (c. 1900-1920)

Worthington proposed the slug as a solution:

The name: "Slug" evokes sluggishness—the tendency of massive objects to resist acceleration (inertia). A more massive object is more "sluggish" in responding to forces.

The definition: 1 slug = mass that accelerates at 1 ft/s² under 1 lbf

The relationship: 1 slug = 32.174 lbm (approximately)

This ratio (32.174) is not arbitrary—it equals the standard acceleration due to gravity in ft/s² (g = 32.174 ft/s²). This means:

• On Earth's surface, a 1-slug mass weighs 32.174 lbf
• On Earth's surface, a 1-lbm mass weighs 1 lbf

Adoption in Engineering Education (1920s-1940s)

The slug gained acceptance in American and British engineering textbooks during the early 20th century:

Advantages recognized:

• Simplified dynamics calculations (F = ma without g_c)
• Clearer distinction between force and mass
• Consistency with scientific notation (separating weight from mass)

Textbook adoption: Engineering mechanics books by authors like Beer & Johnston, Meriam & Kraige, and Hibbeler introduced the slug to generations of engineering students

University courses: American aerospace and mechanical engineering programs taught dynamics using the FPS system with slugs

Aerospace Era Embrace (1940s-1970s)

The slug became essential in American aerospace during the mid-20th century:

NACA/NASA adoption (1940s-1970s):

• Aircraft performance calculations used slugs for mass
• Rocket dynamics required precise force-mass-acceleration relationships
• Apollo program documentation used slugs extensively

Military ballistics:

• Artillery trajectory calculations
• Rocket and missile design
• Aircraft carrier catapult systems

Engineering standards:

• ASME and SAE specifications sometimes used slugs
• Aerospace contractor documentation (Boeing, Lockheed, etc.)

Decline with Metrication (1960s-Present)

Despite its technical superiority, the slug declined for several reasons:

International metrication (1960s onward):

• Most countries adopted SI units (kilogram for mass, newton for force)
• International aerospace and scientific collaboration required metric
• Slug never gained traction outside English-speaking countries

Everyday unfamiliarity:

• People use pounds (lbm/lbf) in daily life, not slugs
• No one says "I weigh 5 slugs" (they say "160 pounds")
• Slug remained a technical unit, never entering popular vocabulary

Educational shifts:

• Even American universities increasingly teach SI units first
• Engineering courses present slugs as "alternative" or "legacy" units

Software standardization:

• Modern engineering software defaults to SI (kg, N, m)
• Maintaining slug support became maintenance burden

Where Slugs Survive Today

The slug persists in specific technical niches:

American aerospace engineering:

• Aircraft weight and balance calculations (sometimes)
• Rocket propulsion dynamics
• Legacy documentation from NASA programs

Mechanical engineering dynamics courses:

• Teaching Newton's laws in FPS units
• Demonstrating unit system consistency

Ballistics and defense:

• Military projectile calculations
• Explosive dynamics

Historical technical documentation:

• 20th-century engineering reports and specifications
• Understanding legacy systems and equipment

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Real-World Examples

Light Objects (0.001-0.1 slugs)

Small everyday items:

• Baseball (5 oz): 0.00967 slugs (~0.145 kg)
• Football (14-15 oz): 0.027 slugs (~0.4 kg)
• Textbook (3 lbs): 0.0932 slugs (~1.36 kg)
• Laptop (4 lbs): 0.124 slugs (~1.81 kg)
• Gallon of water (8.34 lbs): 0.259 slugs (~3.78 kg)

Medium Objects (0.1-1 slug)

Typical human-scale objects:

• Bowling ball (12-16 lbs): 0.373-0.497 slugs (~5.4-7.3 kg)
• Car tire (20-30 lbs): 0.622-0.933 slugs (~9-14 kg)
• Microwave oven (30-50 lbs): 0.933-1.55 slugs (~14-23 kg)
• Average 10-year-old child (70 lbs): 2.18 slugs (~32 kg)

Heavy Objects (1-10 slugs)

Larger items and adult humans:

• Average adult female (160 lbs): 4.97 slugs (~73 kg)
• Average adult male (200 lbs): 6.22 slugs (~91 kg)
• Large dog (80-100 lbs): 2.49-3.11 slugs (~36-45 kg)
• Washing machine (150-200 lbs): 4.66-6.22 slugs (~68-91 kg)
• Motorcycle (400-600 lbs): 12.4-18.7 slugs (~180-270 kg)

Very Heavy Objects (10-100 slugs)

Vehicles and machinery:

• Small car (2,500 lbs): 77.7 slugs (~1,134 kg)
• Midsize car (3,500 lbs): 109 slugs (~1,588 kg)
• Pickup truck (5,000 lbs): 155 slugs (~2,268 kg)
• Large SUV (6,000 lbs): 186 slugs (~2,722 kg)

Extremely Heavy Objects (100+ slugs)

Large vehicles and structures:

• Cement mixer truck (20,000 lbs): 622 slugs (~9,072 kg)
• School bus (30,000 lbs empty): 932 slugs (~13,608 kg)
• Fire truck (40,000 lbs): 1,243 slugs (~18,144 kg)
• Loaded semi-truck (80,000 lbs max): 2,487 slugs (~36,288 kg)

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Common Uses

1. Aerospace Engineering and Aircraft Dynamics

Aerospace engineers use slugs when working in imperial units for aircraft and spacecraft calculations:

Aircraft weight and balance:

• Empty weight: 100,000 lbs = 3,108 slugs
• Loaded weight: 175,000 lbs = 5,440 slugs
• Center of gravity calculations using slugs for mass distribution

Rocket dynamics (Newton's F = ma):

• Thrust: 750,000 lbf
• Mass: 50,000 slugs (initial), decreasing as fuel burns
• Acceleration: F/m = 750,000 lbf / 50,000 slugs = 15 ft/s²

Orbital mechanics:

• Satellite mass in slugs
• Thrust-to-weight calculations
• Momentum and angular momentum in slug·ft/s units

2. Mechanical Engineering Dynamics

Engineering students and professionals analyze motion using slugs:

Newton's second law problems:

• Force: 50 lbf
• Acceleration: 10 ft/s²
• Mass: F/a = 50/10 = 5 slugs (no gravitational constant needed!)

Momentum calculations (p = mv):

• Car mass: 77.7 slugs (2,500 lbs)
• Velocity: 60 ft/s
• Momentum: p = 77.7 × 60 = 4,662 slug·ft/s

Rotational dynamics (moment of inertia):

• I = mr² (with mass in slugs, radius in feet)
• Flywheel: mass = 10 slugs, radius = 2 ft
• I = 10 × 2² = 40 slug·ft²

3. Ballistics and Projectile Motion

Military and firearms engineers use slugs for projectile calculations:

Artillery shell trajectory:

• Shell mass: 0.932 slugs (30 lbs)
• Muzzle force: 50,000 lbf
• Acceleration: a = F/m = 50,000/0.932 = 53,648 ft/s²

Bullet dynamics:

• Bullet mass: 0.000466 slug (150 grains = 0.0214 lbm)
• Chamber pressure force: 0.5 lbf (approximate average)
• Barrel acceleration calculation

Recoil analysis:

• Conservation of momentum (m_gun × v_gun = m_bullet × v_bullet)
• Gun mass: 6.22 slugs (200 lbs)
• Calculating recoil velocity in ft/s

4. Physics Education and Problem Sets

High school and college physics courses teaching imperial units:

Demonstrating unit consistency:

• Showing that F = ma works directly with slugs
• Contrasting with the g_c requirement when using lbm

Inclined plane problems:

• Block mass: 2 slugs
• Angle: 30°
• Friction force calculations in lbf

Atwood machine:

• Two masses in slugs
• Pulley system acceleration
• Tension forces in lbf

5. Automotive Engineering

Vehicle dynamics calculations using imperial units:

Braking force analysis:

• Car mass: 93.2 slugs (3,000 lbs)
• Deceleration: 20 ft/s² (emergency braking)
• Required braking force: F = ma = 93.2 × 20 = 1,864 lbf

Acceleration performance:

• Engine force (at wheels): 3,000 lbf
• Car mass: 77.7 slugs (2,500 lbs)
• Acceleration: a = F/m = 3,000/77.7 = 38.6 ft/s²

Suspension design:

• Spring force (F = kx) in lbf
• Sprung mass in slugs
• Natural frequency calculations

6. Structural Dynamics and Vibration

Engineers analyzing oscillating systems in imperial units:

Simple harmonic motion:

• F = -kx (Hooke's law, force in lbf)
• m = mass in slugs
• Natural frequency: ω = √(k/m) where m is in slugs

Seismic analysis:

• Building mass: distributed load in slugs per floor
• Earthquake force (F = ma) with acceleration in ft/s²

Mechanical vibrations:

• Damping force proportional to velocity
• Mass-spring-damper systems with m in slugs

7. Fluid Dynamics and Hydraulics

Flow and pressure calculations when using imperial units:

Momentum of flowing fluid:

• Mass flow rate: ṁ = ρAv (density in slug/ft³, area in ft², velocity in ft/s)
• Force on pipe bend: F = ṁΔv (in lbf)

Pipe flow:

• Water density: 1.938 slug/ft³ (at 68°F)
• Pressure drop calculations
• Pump power requirements

Aerodynamic forces:

• Drag force (lbf) = ½ ρ v² A C_D
• Air density: 0.00238 slug/ft³ (sea level, standard conditions)

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Conversion Guide

Converting Slugs to Common Units

To Kilograms

Formula: slugs × 14.5939 = kilograms

Examples:

• 1 slug = 1 × 14.5939 = 14.5939 kg
• 5 slugs = 5 × 14.5939 = 72.97 kg
• 100 slugs = 100 × 14.5939 = 1,459.39 kg

To Pounds-Mass

Formula: slugs × 32.174 = pounds-mass

Examples:

• 1 slug = 1 × 32.174 = 32.174 lbm
• 2 slugs = 2 × 32.174 = 64.348 lbm
• 10 slugs = 10 × 32.174 = 321.74 lbm

Why 32.174? This is the standard gravitational acceleration in ft/s² (g = 32.174 ft/s²)

To Grams

Formula: slugs × 14,593.9 = grams

Examples:

• 0.1 slug = 0.1 × 14,593.9 = 1,459.39 g
• 0.5 slug = 0.5 × 14,593.9 = 7,296.95 g

Converting Common Units to Slugs

From Kilograms

Formula: kilograms ÷ 14.5939 = slugs

Examples:

• 10 kg = 10 ÷ 14.5939 = 0.685 slugs
• 70 kg (typical adult) = 70 ÷ 14.5939 = 4.80 slugs
• 1,000 kg = 1,000 ÷ 14.5939 = 68.5 slugs

Quick approximation: kg × 0.0685 ≈ slugs

From Pounds-Mass

Formula: pounds-mass ÷ 32.174 = slugs

Examples:

• 32.174 lbm = 32.174 ÷ 32.174 = 1 slug
• 160 lbm (typical adult) = 160 ÷ 32.174 = 4.97 slugs
• 2,500 lbm (car) = 2,500 ÷ 32.174 = 77.7 slugs

Quick approximation: lbm × 0.0311 ≈ slugs (or divide by ~32)

From Grams

Formula: grams ÷ 14,593.9 = slugs

Examples:

• 1,000 g (1 kg) = 1,000 ÷ 14,593.9 = 0.0685 slugs
• 500 g = 500 ÷ 14,593.9 = 0.0343 slugs

Force-Weight-Mass Relationships

CRITICAL DISTINCTION: Mass vs. Weight

Mass (slugs or lbm):

• Quantity of matter
• Same everywhere in the universe
• Measured with a balance scale (compares to reference mass)

Weight (lbf or newtons):

• Force due to gravity
• Changes with gravitational field strength
• Measured with a spring scale (measures force)

On Earth (g = 32.174 ft/s²):

• 1 slug weighs 32.174 lbf
• 1 lbm weighs 1 lbf
• Weight (lbf) = Mass (slugs) × 32.174

Example:

• Car mass: 77.7 slugs
• Weight on Earth: 77.7 × 32.174 = 2,500 lbf (same numerical value as 2,500 lbm!)
• Weight on Moon (g ≈ 5.32 ft/s²): 77.7 × 5.32 = 413 lbf

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Common Conversion Mistakes

1. Confusing Pounds-Mass with Pounds-Force

The Mistake: Treating lbm and lbf as interchangeable

Why It Happens: On Earth's surface, they have the same numerical value (1 lbm weighs 1 lbf), creating the illusion they're the same

The Truth:

• lbm: Unit of mass (quantity of matter)
• lbf: Unit of force (weight is a force)
• They're only numerically equal on Earth at sea level

Example showing the difference:

• Object mass: 10 lbm = 0.311 slugs
• Weight on Earth: 10 lbf
• Weight on Moon: 10 lbm × (g_moon/g_earth) = 10 × (1/6) = 1.67 lbf (not 10 lbf!)

Correct approach: Always distinguish mass from weight; use slugs to avoid confusion

2. Forgetting the g_c Constant When Using Pounds-Mass

The Mistake: Writing F = ma when using lbm for mass and lbf for force

Why It Happens: Thinking Newton's law always has this simple form

The Truth: If using lbm and lbf: F = ma / g_c

where g_c = 32.174 lbm·ft/(lbf·s²)

Example:

• Mass: 64.348 lbm
• Acceleration: 10 ft/s²
• Wrong: F = 64.348 × 10 = 643.48 lbf ✗
• Correct: F = (64.348 × 10) / 32.174 = 20 lbf ✓

Slug solution: Use slugs (2 slugs = 64.348 lbm) F = ma = 2 × 10 = 20 lbf ✓ (no g_c needed!)

3. Mixing Unit Systems in Calculations

The Mistake: Combining slugs with metric units or using inconsistent units

Why It Happens: Working with mixed-unit data sources

Example of error:

• Mass: 5 slugs
• Acceleration: 9.8 m/s² (metric!)
• F = 5 × 9.8 = 49 ??? (meaningless mixture of units!)

Correct approach: Convert everything to one system first

• Mass: 5 slugs = 72.97 kg
• Acceleration: 9.8 m/s²
• F = 72.97 × 9.8 = 715 N (all metric) ✓

Or:

• Mass: 5 slugs
• Acceleration: 9.8 m/s² = 32.15 ft/s²
• F = 5 × 32.15 = 160.8 lbf (all FPS) ✓

4. Incorrect Slug-to-Kilogram Conversion

The Mistake: Using wrong conversion factors like "1 slug ≈ 15 kg"

Why It Happens: Rounding errors or misremembering

The Truth: 1 slug = 14.5939 kg (exact, based on defined relationships)

Impact on precision:

• Using 15 kg: 5 slugs = 75 kg (error: +2.8%)
• Using 14.59 kg: 5 slugs = 72.95 kg (close, error: -0.03%)
• Correct: 5 slugs = 72.97 kg

Correct approach: Use 14.5939 kg for precision work

5. Confusing Slugs with "Stones" (British Weight Unit)

The Mistake: Thinking slugs are related to stones (another imperial mass unit)

Why It Happens: Both are unfamiliar, archaic-sounding imperial units

The Truth:

• Slug: 32.174 lbm = 14.594 kg (physics/engineering unit)
• Stone: 14 lbm = 6.35 kg (British body weight unit, completely different!)
• No simple relationship between them

Correct understanding: Slugs and stones are unrelated units from different contexts

6. Weight vs. Mass Language Errors

The Mistake: Saying "the object weighs 5 slugs"

Why It Happens: Everyday language uses "weight" for mass

The Truth:

• Slugs measure mass, not weight
• Say: "The object has a mass of 5 slugs"
• The object's weight is 5 slugs × 32.174 ft/s² = 160.87 lbf

Correct usage:

• "This car has a mass of 77.7 slugs" ✓
• "This car weighs 2,500 pounds-force" ✓
• "This car weighs 77.7 slugs" ✗ (weight is measured in lbf, not slugs)

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