Chapter Review

Vectors and Equilibrium

Vector Components and Products · Torque and Equilibrium

Scalars vs Vectors

Scalars carry only magnitude; vectors require both magnitude and direction. All vector algebra (addition, resolution) is geometric, not algebraic.

Key Points

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Scalars: mass, temperature, speed, energy — added algebraically
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Vectors: displacement, velocity, force, acceleration — added geometrically (head-to-tail)
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A null vector ($\vec{0}$) has zero magnitude with undefined direction
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Scalar multiplication by $n$: magnitude scales by $|n|$, direction flips if $n < 0$
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Multiplying a vector by a dimensional scalar produces a new physical quantity (e.g. $m\vec{v}$ = momentum)

Vector Addition and the Resultant

The resultant combines multiple vectors into one equivalent vector. The head-to-tail method and parallelogram law are two geometric constructions for the same result.

Key Points

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Resultant components: $R_x = A_x + B_x$, $R_y = A_y + B_y$
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Resultant magnitude range: $|A - B| \leq R \leq A + B$
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Maximum $R$ when vectors are parallel; minimum when antiparallel
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Addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
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Subtraction defined as adding a negative: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$
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Quadrant of $\vec{R}$ 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 split into perpendicular x and y components using trigonometry. This is the reverse of vector addition and is essential for solving 2D force problems.

Key Points

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$A_x = A\cos\theta$ (horizontal), $A_y = A\sin\theta$ (vertical)
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Magnitude recovery: $A = \sqrt{A_x^2 + A_y^2}$ (Pythagorean theorem)
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Direction recovery: $\theta = \tan^{-1}(A_y / A_x)$ — verify quadrant from component signs
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At $\theta = 45°$: $A_x = A_y = \frac{A}{\sqrt{2}}$
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3D extension: $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$, $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$
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Calculator $\tan^{-1}$ gives $-90°$ to $90°$ only — add $180°$ if $A_x < 0$

Formula

$$A_x = A\cos\theta, \quad A_y = A\sin\theta$$

Scalar (Dot) Product

The dot product yields a scalar measuring how aligned two vectors are. It equals zero for perpendicular vectors — the key orthogonality test.

Key Points

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Angle form: $\vec{A} \cdot \vec{B} = AB\cos\theta$
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Component form: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$
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Commutative: order does not matter
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Self dot product: $\vec{A} \cdot \vec{A} = A^2$
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Positive result → acute angle ($\theta < 90°$); negative → obtuse ($\theta > 90°$)
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Physical application: work $W = \vec{F} \cdot \vec{d}$
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Unit vectors: $\hat{i} \cdot \hat{j} = 0$, $\hat{i} \cdot \hat{i} = 1$

Formula

$$\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$$

Vector (Cross) Product

The cross product yields a vector perpendicular to both inputs. Its magnitude equals the area of the parallelogram formed by the two vectors.

Key Points

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Magnitude: $|\vec{A} \times \vec{B}| = AB\sin\theta$ — maximum at $\theta = 90°$, zero when parallel
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Anti-commutative: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$ — order flips direction
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Direction: right-hand rule — curl fingers from $\vec{A}$ to $\vec{B}$, thumb gives result
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Parallel vectors always give zero cross product (null vector)
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Cyclic unit vectors: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$
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Physical applications: torque $\vec{\tau} = \vec{r} \times \vec{F}$, magnetic force $\vec{F} = q(\vec{v} \times \vec{B})$

Formula

$$\vec{A} \times \vec{B} = AB\sin\theta\,\hat{n}$$

Torque: The Turning Effect

Torque measures a force's ability to produce rotation about a pivot. It depends on force magnitude, distance from pivot, and the angle of application.

Key Points

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$\tau = rF\sin\theta$ — maximum when $\theta = 90°$ (perpendicular), zero when $\theta = 0°$ (force through pivot)
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Moment arm: the perpendicular distance from pivot to the line of action of force, $l = r\sin\theta$
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Vector form: $\vec{\tau} = \vec{r} \times \vec{F}$, direction via right-hand rule
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Torque has units Nm — same dimensions as energy (J) but physically distinct
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Two equivalent views: decompose force perpendicular to $\vec{r}$, or decompose $\vec{r}$ perpendicular to $\vec{F}$

Formula

$$\tau = rF\sin\theta$$

Conditions of Equilibrium

A body is in complete equilibrium when it has zero net force (no translation) and zero net torque (no rotation). Both conditions must be satisfied independently.

Key Points

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First condition: $\sum \vec{F} = 0$ — translational equilibrium
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Second condition: $\sum \vec{\tau} = 0$ — rotational equilibrium
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A couple ($\sum F = 0$, $\sum \tau \neq 0$) shows why both conditions are needed
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Principle of moments: clockwise torques = counter-clockwise torques
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If in equilibrium, net torque is zero about any arbitrary axis
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Smart pivot choice: place it where unknown forces act to eliminate their torques
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Sign convention: counter-clockwise positive, clockwise negative

Formula

$$\sum \vec{F} = 0, \quad \sum \vec{\tau} = 0$$

Center of Gravity and Stability

The center of gravity (CG) is where the entire weight appears to act. Stability depends on whether the CG rises, falls, or stays level when the object is displaced.

Key Points

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$x_{cg} = \frac{\sum w_i x_i}{\sum w_i}$ — weighted average position of all masses
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Stable equilibrium: displacement raises CG → restoring torque returns object
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Unstable equilibrium: displacement lowers CG → object topples further
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Neutral equilibrium: CG stays at same height → object stays in new position
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Uniform bodies: CG at geometric center
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Object remains stable 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 is two equal, opposite, parallel forces with different lines of action. It produces rotation without translation since the net force is always zero.

Key Points

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Couple torque: $\tau = F \times d$ where $d$ is the perpendicular distance between forces
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Net force always zero — causes pure rotation, no translation
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Torque of a couple is the same about any point in the plane
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Increasing $d$ linearly increases torque while keeping net force zero
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Physical example: turning a steering wheel with both hands

Formula

$$\tau_{couple} = F \times d$$

Formulas

Vector Magnitude

Pythagorean length from rectangular components.

Formula

$$|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$$

Rectangular Components

Resolve a vector into perpendicular x and y components.

Formula

$$A_x = A\cos\theta, \quad A_y = A\sin\theta$$

Dot Product

Scalar product — measures alignment; zero means perpendicular.

Formula

$$\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$$

Cross Product

Vector product — result perpendicular to both inputs.

Formula

$$\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$$

Torque

Turning effect — force × lever arm × sine of angle.

Formula

$$\tau = rF\sin\theta$$

Torque Vector Form

Cross product of position vector and force vector.

Formula

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Couple Torque

One force times perpendicular separation between the two forces.

Formula

$$\tau_{couple} = F \times d$$

Center of Gravity

Weighted average position of all constituent weights.

Formula

$$x_{cg} = \frac{\sum w_i x_i}{\sum w_i}$$