Study guide: Engineering Mechanics Fundamentals

📋 Course Outline

1. Fundamental Mechanics Concepts
2. Basic Quantities and SI Units
3. Newton’s Laws and Gravitation
4. Engineering Mechanics Models
5. Scalars Vectors and Force Addition
6. Position and Vector Products
7. Moments and Torque

📖 1. Fundamental Mechanics Concepts

🔑 Key Concepts & Definitions

• Particle: A body whose mass is considered concentrated at a point because its size is negligible.
• Rigid body: A combination of particles whose shape and size are treated as unchanged, although it may sometimes be modeled as a particle with its weight acting at its center of mass.
• Concentrated force: A single point load or a load applied over a small area.

📝 Essential Points

📌 Statics studies how an object remains in equilibrium, whereas dynamics studies why and how an object follows a motion.

💡 Memory Hook

Statics asks how equilibrium is maintained, whereas dynamics asks why and how motion occurs.

📖 2. Basic Quantities and SI Units

🔑 Key Concepts & Definitions

• Mass: A measure of the inertia of a body.
• Force: A vector quantity used to maintain or alter the motion of a body.

★ Must-know

• The SI base units used in mechanics include meters for length, seconds for time, and kilograms for mass.

• The newton is the unit of force, with $1\,\mathrm{N}=1\,\mathrm{kg}\cdot\mathrm{m}/\mathrm{s}^{2}$, so one newton gives a one-kilogram mass an acceleration of one meter per second squared.

Further detail

• The fundamental forces are:

  • gravitational force
  • weak nuclear force
  • electromagnetic force
  • strong nuclear force

• The SI prefixes are giga (G, $10^{9}$), mega (M, $10^{6}$), kilo (k, $10^{3}$), milli (m, $10^{-3}$), micro (μ, $10^{-6}$), and nano (n, $10^{-9}$).

💡 Memory Hook

LTMF: Length, Time, Mass, Force

📖 3. Newton’s Laws and Gravitation

★ Must-know

• Newton’s first law states that a particle at rest or moving with constant velocity remains in that state provided it is not subjected to an unbalanced force.

• Newton’s second law states that a resultant force acting on a particle of mass m produces an acceleration whose magnitude is directly proportional to the force.

• Newton’s third law states that the mutual action and reaction forces between two particles are equal, opposite, and collinear.

Further detail

• The gravitational force on an object of mass m located near Earth is called its weight and depends on the masses of the object and Earth and on the distance measured between their centers.

💡 Memory Hook

Inertia → acceleration → action–reaction

📖 4. Engineering Mechanics Models

★ Must-know

• Engineering mechanics includes:

  • rigid-body mechanics
  • deformable-body mechanics
  • fluid mechanics

• Solid mechanics studies how an object deforms, whereas fluid mechanics studies how a fluid flows.

Further detail

• Rigid-body mechanics includes:
  • translation
  • rotation
  • general motion

💡 Memory Hook

Rigid bodies preserve shape, deformable bodies change shape, and fluids flow.

📖 5. Scalars Vectors and Force Addition

🔑 Key Concepts & Definitions

• Vector: A quantity associated with both magnitude and direction, such as force, weight, displacement, or velocity.

★ Must-know

📌 A scalar is a quantity associated only with magnitude, such as length, mass, position, speed, or time.

• A two-dimensional vector is represented by Cartesian components along the x and y axes, each accompanied by the unit vectors i and j.

Further detail

• Perpendicular vectors are added by forming a right triangle and using Pythagoras’ theorem to determine the resultant magnitude and direction.

📌 Vector addition and subtraction can only be performed on vectors of the same type.

💡 Memory Hook

Represent → resolve → add → subtract

📖 6. Position and Vector Products

🔑 Key Concepts & Definitions

• Unit vector: A dimensionless vector with magnitude one that expresses a direction in a coordinate system.

• Position vector: Connects two given positions in a Cartesian system, and its magnitude gives the distance between the two points.

★ Must-know

📌 The dot product determines a vector’s component parallel or perpendicular to a line and the angle between two vectors, producing a scalar.

Further detail

• The Cartesian unit vectors i, j, and k represent the coordinate directions, and a vector -A has the same magnitude as A but the opposite direction.

• A force directed along a line is represented as a Cartesian vector by multiplying its magnitude by the unit vector along that line.

💡 Memory Hook

Dot product gives projection and a scalar, whereas cross product gives rotation and a vector.

📖 7. Moments and Torque

🔑 Key Concepts & Definitions

• Cross product: Combines two vectors to produce a vector perpendicular to their plane, with magnitude equal to the product of their magnitudes and the sine of the angle between them.

★ Must-know

📐 Formula - The moment or torque of a force about a point is $M=Fd$, where F is the force and d is the perpendicular moment arm.

📌 The direction of a cross product and a three-dimensional moment is determined by the right-hand rule.

📌 The dot product uses cosine and gives a scalar for projection and parallel alignment, whereas the cross product uses sine and gives a vector for perpendicular area and rotational tendency.

Further detail

• A smaller moment arm requires more force to generate the same moment about a point.

💡 Memory Hook

Force acting at a perpendicular distance causes a moment and therefore a rotational tendency.

📊 Synthesis Tables

Dot and Cross Products

Property	Dot product	Cross product
Output	Scalar	Vector
Main use	Projection and parallel alignment	Moments and perpendicular direction
Trigonometric function	Cosine	Sine
Direction rule	None	Right-hand rule