Chapter Review
Vectors and Equilibrium
Vector Analysis · Torque and Equilibrium
Scalars, Vectors, and Unit Vectors
Scalars have magnitude only; vectors require both magnitude and direction. A unit vector ($\hat{A} = \vec{A}/|A|$) extracts pure direction with magnitude 1.
Key Points
- •Vectors are represented as arrows — length = magnitude, tip = direction
- •Two vectors are equal only if both magnitude AND direction match
- •Scalar multiplication by $n$ scales magnitude by $|n|$; negative $n$ reverses direction
- •Standard unit vectors: $\hat{i}$, $\hat{j}$, $\hat{k}$ along x, y, z axes
- •A null vector has zero magnitude and undefined direction
Vector Addition and Subtraction
Vectors combine geometrically via head-to-tail or parallelogram methods. The resultant is the single vector equivalent to all components acting together.
Key Points
- •Component method: $\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$
- •Resultant magnitude bounded: $|A - B| \leq R \leq A + B$
- •Subtraction is adding the negative: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
- •Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
- •Quadrant of resultant determined by signs of $R_x$ and $R_y$
Formula
$$\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$$
Resolution into Rectangular Components
Any vector can be decomposed into perpendicular components along coordinate axes using trigonometry, enabling algebraic treatment of vector problems.
Key Points
- •$A_x = A\cos\theta$, $A_y = A\sin\theta$ where $\theta$ is measured from the positive x-axis
- •Magnitude recovery: $A = \sqrt{A_x^2 + A_y^2}$
- •Direction recovery: $\theta = \tan^{-1}(A_y / A_x)$ — must check quadrant from component signs
- •3D extension: $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$, $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$
- •Calculator gives $\tan^{-1}$ in range $-90°$ to $90°$ only — add $180°$ if $A_x < 0$
Formula
$$A_x = A\cos\theta, \quad A_y = A\sin\theta$$
Dot Product (Scalar Product)
The dot product measures vector alignment, yielding a scalar. It equals zero for perpendicular vectors and is central to calculating work.
Key Points
- •Angle form: $\vec{A} \cdot \vec{B} = AB\cos\theta$
- •Component form: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$
- •Zero when perpendicular ($\theta = 90°$) — the orthogonality test
- •Commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
- •Self dot product: $\vec{A} \cdot \vec{A} = A^2$
- •Physical application: Work $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$
Formula
$$\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$$
Cross Product (Vector Product)
The cross product yields a vector perpendicular to both inputs, with magnitude $AB\sin\theta$. Direction follows the right-hand rule.
Key Points
- •Anti-commutative: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$ — order matters
- •Zero for parallel/antiparallel vectors ($\sin 0° = \sin 180° = 0$)
- •Maximum when perpendicular: $|\vec{A} \times \vec{B}| = AB$
- •Cyclic unit vector products: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$
- •Geometric meaning: magnitude equals the area of the parallelogram formed by the two vectors
- •Physical applications: torque ($\vec{r} \times \vec{F}$), magnetic force ($q\vec{v} \times \vec{B}$)
Formula
$$\vec{A} \times \vec{B} = AB\sin\theta\,\hat{n}$$
Position Vectors and Displacement
A position vector locates a point relative to the origin. The displacement between two points is the difference of their position vectors.
Key Points
- •$\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$ for point P(a, b, c)
- •Distance from origin: $r = \sqrt{a^2 + b^2 + c^2}$
- •Displacement: $\vec{AB} = \vec{r}_B - \vec{r}_A$
- •Distance between points: $|AB| = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2}$
Formula
$$\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$$
Torque (Moment of a Force)
Torque measures a force's ability to cause rotation about a pivot. It depends on force magnitude, distance from pivot, and the angle of application.
Key Points
- •$\tau = rF\sin\theta$ — maximum at $\theta = 90°$, zero when force passes through pivot
- •Moment arm: perpendicular distance from pivot to line of action ($l = r\sin\theta$)
- •Vector form: $\vec{\tau} = \vec{r} \times \vec{F}$, direction via right-hand rule
- •Units: N·m (same dimensions as energy but physically distinct)
- •Doubling the lever arm doubles the torque for the same force
Formula
$$\tau = rF\sin\theta$$
Conditions of Equilibrium
Static equilibrium requires both zero net force (no translation) and zero net torque (no rotation). Both conditions must be satisfied simultaneously.
Key Points
- •First condition: $\sum \vec{F} = 0$ (translational equilibrium)
- •Second condition: $\sum \vec{\tau} = 0$ (rotational equilibrium)
- •Principle of moments: $\sum \tau_{cw} = \sum \tau_{ccw}$
- •If net torque is zero about one point, it is zero about ANY point
- •Smart pivot choice: place pivot where unknown forces act to eliminate them from the torque equation
- •A negative force result means the assumed direction was wrong — flip it, don't discard it
Formula
$$\sum \vec{F} = 0, \quad \sum \vec{\tau} = 0$$
Center of Gravity and Stability
The center of gravity is the effective point where total weight acts. Its position relative to the base of support determines an object's stability type.
Key Points
- •CG position: $x_{cg} = \frac{\sum w_i x_i}{\sum w_i}$ (weighted average)
- •Stable equilibrium: displacement raises CG → restoring torque returns object
- •Unstable equilibrium: displacement lowers CG → object topples further
- •Neutral equilibrium: CG height unchanged → object stays in new position
- •Uniform bodies: CG at geometric center
- •Object remains upright as long as vertical line from CG falls within the base of support
Formula
$$x_{cg} = \frac{\sum w_i x_i}{\sum w_i}$$
Couples and Pure Rotation
A couple consists of two equal, opposite, parallel forces with different lines of action, producing pure rotation with zero net force.
Key Points
- •Torque of a couple: $\tau = F \times d$ where $d$ is the perpendicular separation
- •Net force is always zero — no translational motion
- •Torque of a couple is the same about ANY point in the plane
- •Increasing the arm $d$ linearly increases the torque
Formula
$$\tau_{couple} = F \times d$$
Formulas
Rectangular Components
Resolve a vector into x and y components using the angle with the x-axis.
Formula
$A_x = A\cos\theta, \quad A_y = A\sin\theta$
Vector Magnitude
Find the length of a vector from its components via the Pythagorean theorem.
Formula
$|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$
Dot Product
Scalar product measuring alignment between two vectors.
Formula
$\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$
Cross Product Magnitude
Magnitude of the vector product; equals parallelogram area.
Formula
$|\vec{A} \times \vec{B}| = AB\sin\theta$
Angle Between Vectors
Find the angle using the dot product divided by the product of magnitudes.
Formula
$\cos\theta = \frac{\vec{A} \cdot \vec{B}}{|A||B|}$
Torque Formula
Turning effect of a force about a pivot point.
Formula
$\tau = rF\sin\theta$
Equilibrium Conditions
Both net force and net torque must be zero for static equilibrium.
Formula
$\sum \vec{F} = 0, \quad \sum \vec{\tau} = 0$
Center of Gravity
Weighted average position of all component weights.
Formula
$x_{cg} = \frac{\sum w_i x_i}{\sum w_i}$