Question

Statement I : If three forces \(\vec{F_1}\), \(\vec{F_2}\) and \(\vec{F_3}\) are represented by three sides of a triangle and \(\vec{F_1} + \vec{F_2} = -\vec{F_3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II : A triangle made up of three forces \(\vec{F_1}\), \(\vec{F_2}\) and \(\vec{F_3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both Statement I and Statement II are true.
• B. Both Statement I and Statement II are false.
• C. Statement I is true but Statement II is false.
• D. Statement I is false but Statement II is true.

Hint:

Lami's Theorem is often used for three concurrent forces in equilibrium: \(\frac{F_1}{\sin \alpha} = \frac{F_2}{\sin \beta} = \frac{F_3}{\sin \gamma}\). Geometrically, if vectors form a closed polygon, the net force is always zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a body to be in translatory equilibrium, the vector sum of all forces acting on it must be zero (\(\sum \vec{F} = 0\)). If three forces are concurrent (acting at the same point) and their vector sum is zero, they keep the particle in equilibrium.
Step 2: Detailed Explanation:
Analysis of Statement I:
The condition given is \(\vec{F_1} + \vec{F_2} = -\vec{F_3}\).
Rearranging this, we get \(\vec{F_1} + \vec{F_2} + \vec{F_3} = 0\).
By the triangle law of vector addition, if three vectors form a closed triangle when placed tail-to-head, their sum is zero.
If these forces are concurrent, this zero net force condition directly implies that the particle is in equilibrium. Thus, Statement I is true.
Analysis of Statement II:
When forces \(\vec{F_1}\), \(\vec{F_2}\), and \(\vec{F_3}\) are taken as sides of a triangle in the same order (cyclic order), the starting point of the first vector coincides with the terminal point of the last vector.
This geometric configuration signifies that the resultant vector is a null vector (\(\vec{R} = 0\)).
Since the net force is zero, the condition for translatory equilibrium is satisfied. Thus, Statement II is true.
Step 3: Final Answer:
Both statements correctly describe the conditions for equilibrium using vector geometry. Therefore, both are true.