Mechanics and Vectors Part 1.1-1.3

Vocabulary practice flashcards covering the fundamentals of mechanics, Newton's laws, and vector algebra principles.

Mechanics

The science studying forces and their effects, including stresses, strains, motion, and deformation.

Statics

The branch of mechanics dealing with objects in equilibrium, which are either stationary or moving at a constant velocity (acceleration $= 0$).

Dynamics

The branch of mechanics dealing with objects in motion or accelerating (acceleration $\neq 0$).

Body Types in Mechanics

The four types of bodies studied in mechanics: mass particle, rigid body, deformable body, and fluid.

Inertial Frame

A reference frame where Newton's laws hold true, characterized by zero resultant force resulting in zero acceleration.

Newton's First Law (Law of Inertia)

The law stating that if the sum of forces acting on an object is zero, then its velocity remains constant.

Newton's Second Law

The law stating that force equals mass times acceleration ($F = ma$) for a body with constant mass.

Newton's Third Law (Action-Reaction)

The law stating that forces occur in pairs; if Object A exerts a force on Object B, Object B exerts an equal and opposite force on Object A, acting on different bodies without canceling each other out.

Universal Gravitation Formula

The formula $F = G \frac{m_1 m_2}{r^2}$, which calculates the gravitational attractive force between two masses.

Weight Formula

The formula $W = mg$, where the SI nominal value of acceleration due to gravity is $g = 9.81\,m/s^2$.

Engineering Accuracy

The standard practice in engineering of consistently keeping $3$ to $4$ significant digits in calculations.

Scalar

A quantity that has magnitude only, represented by a real number and a unit (e.g., mass, time).

Vector

A quantity that has a non-negative magnitude, a unit, and a direction (e.g., force, velocity).

Unit Vector

A vector that has a magnitude (length) of exactly $1$.

Directional Requirements for a 2D Vector

The parameters needed to define a two-dimensional vector: its magnitude plus $1$ independent angle, or $2$ components ($u_x, u_y$).

Directional Requirements for a 3D Vector

The parameters needed to define a three-dimensional vector: its magnitude plus $2$ independent angles, or $3$ components ($u_x, u_y, u_z$).

Scalar Multiplication of Vectors

Multiplying a vector $\mathbf{u}$ by a scalar $\alpha$ to produce a parallel vector with magnitude scaled by $\alpha$, pointing in the same direction if $\alpha > 0$ and the opposite direction if $\alpha < 0$.

Vector Representation via Unit Vector

The expression showing that any vector can be written as the product of its magnitude and its unit vector: $\mathbf{u} = u \cdot \mathbf{u}_{\text{unit}}$.

Properties of scalar multiplication of Vectors

• Associative: α(βu) = (αβ)u

• Distributive (Scalar Addition): (α + β)u = αu + βu

• Distributive (Vector Addition): α(u + v) = αu + αv

Vector Addition Geometric Rules

The geometric methods used to add vectors, specifically the Parallelogram Rule and the Triangle Rule (forming triangle ABC).

Vector Magnitude Inequality

The principle stating that $|\mathbf{u} + \mathbf{v}| \neq |\mathbf{u}| + |\mathbf{v}|$, (Equality holds only if u and v are perfectly aligned/collinear; otherwise, the triangle inequality applies).

Properties of vector addition

Properties:

• Commutative: u + v = v + u (Order does not matter).

• Associative: (u + v) + w = u + (v + w)

Diagonals of a Parallelogram (Vector Addition)

In the Parallelogram Rule, the first diagonal represents vector sum ($\mathbf{u} + \mathbf{v}$), and the second diagonal represents vector difference ($\mathbf{u} - \mathbf{v}$ or $\mathbf{v} - \mathbf{u}$).