Fundamentals of Mechanical Engineering

Engineering Units & Systems

The SI System (International System)

Mechanical engineers primarily use SI units (metric system). Understanding units is critical: many engineering failures come from unit errors.

Base SI Units for Mechanical Engineering

Derived Units

Common Prefixes

US Customary Units (Still Common)

Many US industries still use Imperial/US units:

Key Conversions

• 1 inch = 25.4 mm
• 1 foot = 0.3048 m
• 1 lbf = 4.448 N
• 1 psi = 6,895 Pa
• 1 hp = 746 W

Unit Conversion Tips

Always use dimensional analysis:

Example: Convert 60 mph to m/s

60 miles/hour × (1 hour/3600 s) × (5280 ft/1 mile) × (0.3048 m/1 ft)
= 60 × (5280 × 0.3048) / 3600
= 26.8 m/s

Pro Tip: Write units in every step of calculations to catch errors!

Forces

What is a Force?

A force is a push or pull that can cause an object to accelerate, deform, or change state. Measured in Newtons (N) or pounds-force (lbf).

Types of Forces

1. Contact Forces

   • Normal force (perpendicular to surface)
   • Friction force (opposes motion)
   • Tension (pulling through rope/cable)
   • Compression (pushing)

2. Body Forces

   • Gravity/Weight (W = m × g)
   • Magnetic force
   • Electrostatic force

3. Applied Forces

   • External loads
   • Pressures
   • Distributed loads

Newton's Laws of Motion

First Law (Inertia)

An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted upon by a net external force.

Second Law (F = ma)

The acceleration of an object is directly proportional to the net force and inversely proportional to its mass.

F = m × a

Where:
- F = net force (N)
- m = mass (kg)
- a = acceleration (m/s²)

Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction.

When you push a wall with 100 N, the wall pushes back on you with 100 N.

Weight vs Mass

Mass (m): Amount of matter, measured in kg
Weight (W): Force due to gravity, W = m × g

On Earth: g = 9.81 m/s² (or approximately 10 m/s²)

Example:
- Mass: 70 kg (same everywhere)
- Weight on Earth: W = 70 kg × 9.81 m/s² = 686.7 N
- Weight on Moon: W = 70 kg × 1.62 m/s² = 113.4 N

Vectors

Forces have both magnitude (size) and direction, making them vectors.

Vector Representation

    ↑ y
    |
    |    F (force vector)
    |   /
    |  /
    | /θ
    |/________→ x

Component Form:

• Fx = F × cos(θ), horizontal component
• Fy = F × sin(θ), vertical component

Magnitude:

• |F| = √(Fx² + Fy²)

Direction:

• θ = tan⁻¹(Fy / Fx)

Vector Addition

To add vectors, add their components:

F1 = (F1x, F1y)
F2 = (F2x, F2y)
Fresultant = (F1x + F2x, F1y + F2y)

Example:

F1 = 50 N at 0° (horizontal)  → (50, 0)
F2 = 30 N at 90° (vertical)   → (0, 30)

Fresultant = (50, 30)
|F| = √(50² + 30²) = √(2500 + 900) = √3400 = 58.3 N
θ = tan⁻¹(30/50) = 31°

3D Vectors

In 3D space, vectors have three components:

F = (Fx, Fy, Fz)

Where:
Fx = F × cos(α)  — x-component
Fy = F × cos(β)  — y-component  
Fz = F × cos(γ)  — z-component

|F| = √(Fx² + Fy² + Fz²)

Free Body Diagrams (FBDs)

Most important skill in mechanics! A Free Body Diagram shows all forces acting on an object.

How to Draw a FBD

1. Isolate the object of interest
2. Draw all forces as vectors
3. Label magnitude and direction
4. Include reactions at supports

Example: Block on a Table

        ↑ N (Normal force from table)
        |
    [========]  ← block (mass m)
        |
        ↓ W = mg (Weight)

Forces on the block:

• Weight W = mg (downward)
• Normal force N (upward from table)

If at rest: N = W (equilibrium)

Example: Block on Inclined Plane

         N ↑ (perpendicular to surface)
           |
           |
       [====] ← block
      /    ↓ W = mg
     /  θ
    /________

Components:

• Weight parallel to plane: W∥ = mg sin(θ)
• Weight perpendicular to plane: W⊥ = mg cos(θ)
• Normal force: N = mg cos(θ)

Common Support Types

Equilibrium

An object is in equilibrium when:

1. ΣF = 0 (sum of all forces = 0)
2. ΣM = 0 (sum of all moments = 0)

In 2D (Planar Problems)

ΣFx = 0  — sum of horizontal forces = 0
ΣFy = 0  — sum of vertical forces = 0
ΣM = 0   — sum of moments = 0

Example Problem: Hanging Sign

     /|\ 
    / | \
   /  |  \
  T1  |  T2  (tension cables at 45° and 30°)
      |
   [SIGN]  W = 200 N

Given: Sign weighs 200 N, T1 at 45°, T2 at 30° from horizontal

Find: T1 and T2

Solution:

ΣFx = 0:  T1 cos(45°) - T2 cos(30°) = 0
ΣFy = 0:  T1 sin(45°) + T2 sin(30°) - 200 = 0

From equation 1:
T1 × 0.707 = T2 × 0.866
T1 = 1.225 T2

Substitute into equation 2:
1.225 T2 × 0.707 + T2 × 0.5 = 200
0.866 T2 + 0.5 T2 = 200
1.366 T2 = 200
T2 = 146.4 N

T1 = 1.225 × 146.4 = 179.3 N

Practice Problems

Problem 1: Unit Conversion

Convert 100 km/h to m/s.

<details> <summary>Solution</summary>

100 km/h × (1000 m / 1 km) × (1 h / 3600 s)
= 100 × 1000 / 3600
= 27.8 m/s

</details>

Problem 2: Vector Addition

Two forces act on a bolt: F1 = 100 N east, F2 = 150 N north. Find the resultant force.

<details> <summary>Solution</summary>

F1 = (100, 0) N
F2 = (0, 150) N
Fresultant = (100, 150) N

Magnitude: |F| = √(100² + 150²) = √(10000 + 22500) = 180.3 N
Direction: θ = tan⁻¹(150/100) = 56.3° from east

</details>

Problem 3: Weight Calculation

What is the weight of a 500 kg machine on Earth?

<details> <summary>Solution</summary>

W = m × g
W = 500 kg × 9.81 m/s²
W = 4,905 N ≈ 4.9 kN

</details>

Key Takeaways

✓ Units are critical: always include them and convert carefully
✓ Forces are vectors: magnitude and direction both matter
✓ FBDs are essential: master drawing them for every problem
✓ Equilibrium means no acceleration (ΣF = 0, ΣM = 0)
✓ Break vectors into components: makes calculations easier

Next Steps

Now that you understand fundamental concepts, you're ready for:

• Chapter 02: Statics: analyzing structures and systems at rest
• Practice drawing FBDs for objects around you
• Work through additional equilibrium problems

Pro Tip: Keep a notebook of FBDs and solutions. Pattern recognition comes with practice!