Scalars, Graphical Vectors and Force Components

Scalars vs. Vector Quantities

All physical quantities in mechanics are classified as either scalars or vectors:

Vector Addition & Rectangular Components

When multiple forces act simultaneously on a single body (concurrent forces), their combined effect is represented by a single resultant vector ($R$).

Vector Addition Laws

1. Parallelogram Law of Forces: If two concurrent forces acting on a point are represented in magnitude and direction by two adjacent sides of a parallelogram, their resultant is represented by the diagonal passing from that point.
2. Triangle Law of Forces: If two forces acting on a point are drawn tip-to-tail, the vector drawn from the initial tail to the final tip represents the resultant force.

Analytical Resolution into Perpendicular Components

In engineering practice, resolving vectors into orthogonal horizontal ($x$) and vertical ($y$) components is much faster and more accurate than graphical scale drawings.

For a force $F$ acting at an angle $\theta$ measured relative to the horizontal axis:

$$F_x = F \cos\theta$$ $$F_y = F \sin\theta$$

To recombine multiple resolved components into a single resultant force $R$:

$$R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$$ $$\theta_R = \arctan\left(\frac{|\sum F_y|}{|\sum F_x|}\right)$$

Rigging & Hoisting Mechanics

When lifting equipment with a multi-leg bridle sling, the tension in each sling leg depends heavily on the sling angle ($\theta$) measured relative to the horizontal surface of the load.

Sling Tension Formula

For a symmetrical two-leg sling lifting a total load weight $W$:

$$T = \frac{W}{2 \sin\theta}$$

Where:

• $T$ = Tension in each sling leg ($\text{N}$ or $\text{kN}$)
• $W$ = Total weight of load ($m \times g$, in $\text{N}$ or $\text{kN}$)
• $\theta$ = Angle between the sling leg and the horizontal load plane

Alternatively, using the ratio of sling length ($L$) to vertical height ($H$ from load to hook):

$$\text{Load Angle Factor (LAF)} = \frac{L}{H} = \frac{1}{\sin\theta}$$ $$T = \left(\frac{W}{2}\right) \times \left(\frac{L}{H}\right)$$

The Critical Danger of Shallow Sling Angles

As the sling angle becomes shallower, $\sin\theta$ approaches zero, causing the internal tension $T$ to increase nonlinearly as the reciprocal of sine:

Crucial Exam Fact: At a $30^\circ$ horizontal sling angle, each sling leg is subjected to a tensile force equal to the entire weight of the load! Use the sling manufacturer’s ratings and the approved lifting plan; this arithmetic does not authorize a lift.

Construct a graphical resultant

Choose a scale such as 1 cm for 2 kN and mark north/east or positive axes. Draw the first force arrow to scale. From its tip draw the next arrow in its correct direction. The resultant runs from the first tail to the final tip. Its length gives magnitude and its measured angle gives direction. An equilibrant has equal magnitude and the opposite direction. Never add arrow lengths as scalars unless the forces share a line of action and their signs have been considered.

For 6 kN east and 8 kN north, draw perpendicular arrows 3 cm and 4 cm long. The closing diagonal measures 5 cm, giving a 10 kN resultant at about 53.1 degrees north of east. Analytical checking gives the same magnitude from the square root of 6 squared plus 8 squared. The inverse tangent supplies only a reference angle; use the signs of both components to select the correct quadrant. A resultant pointing southwest cannot be described by the northeast reference angle alone.

In a symmetric two-leg lifting calculation, state that both legs share the load equally, the load is stationary, and the angle is measured from the horizontal. The simple tension equation does not establish the capacity of hooks, shackles, attachment points or the load itself. Unequal leg lengths or a displaced centre of gravity invalidate equal sharing. Four sling legs do not automatically divide a load into four equal parts. The exam tests vector resolution; actual rigging additionally requires rated equipment and competent lift planning.

Source notes

Official scope and technical references checked 2026-10-17: Reference 1; Reference 2. Archived DOE handbooks are used for physical principles, not current Canadian legal requirements.