Forces and Equilibrium: Question 1

Worked solution

Step 1: Set up the two known forces as perpendicular components

Take east as the positive $x$-direction and north as the positive $y$-direction. The two known forces are already perpendicular to each other: $F_1 = 9\text{ N (east)}, \qquad F_2 = 12\text{ N (north)}$

Step 2: Find the resultant of these two forces

Since the point is in equilibrium, the three forces sum to zero. This means the unknown force $F$ must be exactly equal in magnitude, and opposite in direction, to the resultant of the $9\text{ N}$ and $12\text{ N}$ forces (drawn tip-to-tail, the three forces form a closed triangle: the “triangle of forces”).

The resultant $R$ of two perpendicular forces is found with Pythagoras: $R = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$

Step 3: State the value of $F$

For equilibrium, $F$ must balance this resultant exactly, so: $F = 15\text{ N}$

(Note $9, 12, 15$ is a scaled-up $3$–$4$–$5$ right-angled triangle. A useful pattern to recognise.)

Why the other options are wrong

• B ($21\text{ N}$): this is just $9+12$, treating the two forces as if they acted in the same direction rather than combining them as perpendicular vectors.
• C ($3\text{ N}$): this is $12-9$, which has no physical meaning here, forces at right angles cannot be combined by simple subtraction.
• D ($10.5\text{ N}$): this is the average of $9$ and $12$, not the magnitude of their vector resultant.

Final answer

• $F = \boxed{15}\text{ N}$, option A.