Question

Three coplanar equilibrium forces \( P = 40\text{ N} \), \( Q = x\text{ N} \) and \( R = y\text{ N} \) are acting on a body and the body is said to be in equilibrium when the angle included between each other is \( 120^\circ \), then \( x = \)

• A. \( 20\text{ N} \)
• B. \( 40\text{ N} \)
• C. \( 60\text{ N} \)
• D. \( 80\text{ N} \)

Hint:

When three concurrent coplanar forces maintain static equilibrium and have completely identical angles of \( 120^\circ \) separating them, the force system is perfectly symmetrical. This symmetry means all three force magnitudes must be exactly equal to each other! Thus, \( P = Q = R = 40\text{ N} \) instantly.

Answer 1

The Correct Option is B

Concept: Lami's Theorem states that if three coplanar forces act at a point and keep it in a state of static equilibrium, then each force is directly proportional to the sine of the angle included between the remaining two forces. Mathematically, let three forces \( P \), \( Q \), and \( R \) act at a single node point. Let \( \alpha \) be the interior angle opposite to force \( P \) (between \( Q \) and \( R \)), let \( \beta \) be the interior angle opposite to force \( Q \) (between \( P \) and \( R \)), and let \( \gamma \) be the interior angle opposite to force \( R \) (between \( P \) and \( Q \)). The mathematical relation is expressed as: \[ \frac{P}{\sin\alpha} = \frac{Q}{\sin\beta} = \frac{R}{\sin\gamma} \]

Step 1: Identifying parameters and given values.
From the statement of the problem, we have three coplanar forces acting in equilibrium:

• Magnitude of the first force, \( P = 40\text{ N} \)

• Magnitude of the second force, \( Q = x\text{ N} \)

• Magnitude of the third force, \( R = y\text{ N} \)
The problem states that the angle included between any two adjacent forces is exactly equal, with a uniform value of \( 120^\circ \). Therefore, the geometric angles opposite to each respective force vector are: \[ \alpha = 120^\circ, \quad \beta = 120^\circ, \quad \gamma = 120^\circ \]

Step 2: Setting up Lami's Theorem ratio relation.
Let us substitute these identified forces and their corresponding opposite angular values directly into Lami's ratio equation: \[ \frac{40}{\sin(120^\circ)} = \frac{x}{\sin(120^\circ)} = \frac{y}{\sin(120^\circ)} \]

Step 3: Solving for the unknown variable \( x \).
To isolate the variable parameter \( x \), we can equate the first ratio component directly to the second ratio component in our chain of equations: \[ \frac{40}{\sin(120^\circ)} = \frac{x}{\sin(120^\circ)} \] Since the denominator term \( \sin(120^\circ) \) appears identically on both the left-hand and right-hand sides of the equality, we can safely multiply both sides of the equation by \( \sin(120^\circ) \) to eliminate it: \[ 40 = x \quad \Rightarrow \quad x = 40\text{ N} \] Hence, the magnitude of the unknown force \( Q \) is precisely calculated to be \( 40\text{ N} \). This aligns with option (B).