Three forces keep a particle in equilibrium. One acts towards east, another towards north-west and the third towards south. If the first force is 5 N, find the other two.

Problem Statement

Particle in equilibrium under three forces

Fig. 2.31: Forces acting on the particle

Step-by-Step Solution

Key Information & Angles

A particle is in equilibrium under three concurrent forces.
Force 1 (R): 5 N, acting towards East (E).
Force 2 (P): Unknown magnitude, acting towards North-West (NW).
Force 3 (Q): Unknown magnitude, acting towards South (S).
Angle between Q (S) and P (NW) = Angle opposite to 5N force = $135^\circ$.
Angle between P (NW) and 5N (E) = Angle opposite to Q force = $135^\circ$.
Angle between 5N (E) and Q (S) = Angle opposite to P force = $90^\circ$.

Step 1: Apply Lami’s Theorem

Lami’s theorem is applicable as there are three concurrent forces acting on a particle in equilibrium. It states that each force is proportional to the sine of the angle between the other two forces.

Applying Lami’s theorem (P is NW force, Q is S force, 5N is E force):

$$ \frac{P}{\sin(90^\circ)} = \frac{Q}{\sin(135^\circ)} = \frac{5}{\sin(135^\circ)} $$

Step 2: Calculate Force P (North-West)

From the Lami’s theorem relation, isolate P:

$$ \frac{P}{\sin(90^\circ)} = \frac{5}{\sin(135^\circ)} $$ $$ P = 5 \times \frac{\sin(90^\circ)}{\sin(135^\circ)} $$

Substitute known trigonometric values ($ \sin(90^\circ) = 1 $ and $ \sin(135^\circ) = \sin(45^\circ) = \frac{1}{\sqrt{2}} $):

$$ P = 5 \times \frac{1}{1/\sqrt{2}} $$ $$ P = 5 \times \sqrt{2} $$ $$ P = 5\sqrt{2} \, \text{N} $$

Approximation: $ 5\sqrt{2} \approx 5 \times 1.414 = 7.07 \, \text{N} $

Step 3: Calculate Force Q (South)

From the Lami’s theorem relation, isolate Q:

$$ \frac{Q}{\sin(135^\circ)} = \frac{5}{\sin(135^\circ)} $$

Since $ \sin(135^\circ) $ appears on both denominators (and is non-zero), we can simplify:

$$ Q = 5 \times \frac{\sin(135^\circ)}{\sin(135^\circ)} $$ $$ Q = 5 \times 1 $$ $$ Q = 5 \, \text{N} $$

Final Result

Force towards North-West (P) = $ \mathbf{5\sqrt{2} \, N} $ ($ \approx 7.07 \, N $)
Force towards South (Q) = $ \mathbf{5 \, N} $

Conceptual Explanation & Applications

Core Concepts:

Static Equilibrium: For a particle to be in equilibrium, the vector sum of all forces acting on it must be zero ($ \sum \vec{F} = 0 $). This means the net force is zero, and the particle does not accelerate.
Concurrent Forces: The three forces act through a single point (the particle).
Lami’s Theorem: A direct application of the sine rule to the triangle of forces for three concurrent forces in equilibrium. It states that the magnitude of each force is proportional to the sine of the angle between the other two forces. $ \frac{P}{\sin \alpha} = \frac{Q}{\sin \beta} = \frac{R}{\sin \gamma} $, where $ \alpha, \beta, \gamma $ are the angles opposite forces P, Q, R respectively.
Vector Directions & Angles: Accurately determining the angles between the forces based on their given directions (East, North-West, South) is crucial for applying Lami’s theorem correctly.

Real-World Applications:

Statics & Structures: Analyzing forces in simple structural elements like cables supporting a weight, simple trusses, or brackets fixed to a wall.
Mechanical Systems: Determining tensions or compressions in linkages or supports where three members meet at a point.
Navigation & Physics: Problems involving forces like wind, current, and thrust acting on an object in equilibrium.
Educational Demonstrations: Used with force tables in physics labs to demonstrate vector addition and equilibrium conditions.

Why It Works:
Since the particle is in equilibrium, the three force vectors must form a closed triangle when added head-to-tail. Lami’s theorem is essentially the sine rule applied to this force triangle. The angles *between* the forces (when arranged tail-to-tail, as in the diagram) are supplementary to the internal angles of the force triangle (e.g., angle between P and Q is $135^\circ$, internal angle opposite 5N in the force triangle is $180^\circ-135^\circ=45^\circ$). Since $ \sin(\theta) = \sin(180^\circ-\theta) $, Lami’s theorem works using the angles between the forces directly. It provides a convenient mathematical relationship to solve for unknown forces when exactly three concurrent forces maintain equilibrium.