Statement I If three forces F1, F2 and F3 are represented by three sides of a triangle and F1 + F2 =

Q: What exactly does 'taken in the same order' mean for vectors?

It means the vectors are arranged head-to-tail continuously. The head of the first vector connects to the tail of the second, and so on, forming a continuous path without any sudden reversals.

Q: Why does a closed triangle of forces guarantee equilibrium?

When vectors form a closed loop, the starting point and the ending point are exactly the same. This means the net displacement—or in this case, the net force—is zero, which is the exact condition for translatory equilibrium.

Q: Can forces form a triangle but not be concurrent?

Yes! If forces act on an extended rigid body, they could form a closed triangle geometrically but act at different points, creating a net torque (a couple). However, for a point particle, all forces act at the same point, so they are inherently concurrent.

Q: What is the difference between translatory and rotational equilibrium?

Translatory equilibrium means the net force is zero ($\mathbf{F}_{\text{net}} = 0$), so the object won't accelerate linearly. Rotational equilibrium means the net torque is zero ($\boldsymbol{\tau}_{\text{net}} = 0$), so it won't accelerate angularly.

Q: Is it possible for $\mathbf{F}_1 + \mathbf{F}_2 = \mathbf{F}_3$ to represent equilibrium?

No! If $\mathbf{F}_1 + \mathbf{F}_2 = \mathbf{F}_3$, then the net force is $\mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = \mathbf{F}_3 + \mathbf{F}_3 = 2\mathbf{F}_3$, which is not zero. For equilibrium, the sum must be exactly zero.

Question

Statement I If three forces $\mathbf{F}_1, \mathbf{F}_2$ and $\mathbf{F}_3$ are represented by three sides of a triangle and $\mathbf{F}_1 + \mathbf{F}_2 = -\mathbf{F}_3$, then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II A triangle made up of three forces $\mathbf{F}_1, \mathbf{F}_2$ and $\mathbf{F}_3$ 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.

Options:

(A) Statement I is false but statement II is true.

(B) Statement I is true but statement II is false.

(C) Both statement I and statement II are false.

(D) Both statement I and statement II are true.

Sigma Percentile · Accepted Answer

### The Elegance of Vector Equilibrium

Imagine you are playing a game of tug-of-war, but instead of two teams pulling in opposite directions, there are three teams pulling from different angles. If the knot in the center of the rope doesn't move an inch, you have just witnessed the beauty of **translatory equilibrium**. This problem tests our fundamental understanding of how vectors behave when a system is perfectly balanced.

### Decoding the Mathematical Condition

Let's break down **Statement I**. We are given a mathematical relationship between three forces:

$$ \mathbf{F}_1 + \mathbf{F}_2 = -\mathbf{F}_3 $$

At first glance, this is just an algebraic equation. But in physics, every equation tells a story. If we bring $\mathbf{F}_3$ to the left side of the equation, we get:

$$ \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = 0 $$

This is the ultimate condition for translatory equilibrium! It tells us that the **net force** acting on the object is exactly zero. According to Newton's Second Law, if the net force is zero, the acceleration is zero, and the object remains at rest (or in uniform motion). If these three forces act simultaneously at a single point, they are called **concurrent forces**. Because their vector sum is zero, they perfectly cancel each other out, keeping the particle in equilibrium. Thus, Statement I is absolutely true.

### The Geometry of Equilibrium

Now, let's look at **Statement II**, which shifts our perspective from algebra to geometry. It states that the three forces form the sides of a triangle, taken in the same continuous order.

What does "taken in the same order" mean? It means we arrange the vectors head-to-tail. The tail of $\mathbf{F}_2$ starts at the head of $\mathbf{F}_1$, and the tail of $\mathbf{F}_3$ starts at the head of $\mathbf{F}_2$. Because they form a closed triangle, the head of the final vector $\mathbf{F}_3$ lands exactly back at the tail of the first vector $\mathbf{F}_1$.

According to the **Polygon Law of Vector Addition**, the resultant of a set of vectors arranged head-to-tail is drawn from the starting tail to the final head. But here, the start and end points are the exact same location! The net displacement is zero, which mathematically means:

$$ \mathbf{F}_{	ext{net}} = \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = 0 $$

Once again, a net force of zero guarantees translatory equilibrium. Statement II is a beautiful geometric restatement of the exact same physical reality.

### The Final Verdict

Both statements are conceptually flawless. Statement I proves equilibrium through algebraic manipulation, while Statement II proves it through geometric vector addition. Therefore, both Statement I and Statement II are true.

Select Answer:

Visualized Solution

$Analyzing the Statements$

$Condition for Equilibrium$

$Concurrent Forces$

$Polygon Law of Vectors$

$Triangle of Forces$

$Conclusion$

The Sigma Insight: Equilibrium of Concurrent Forces

The Elegance of Vector Equilibrium

Decoding the Mathematical Condition

The Geometry of Equilibrium

The Final Verdict

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Statements

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Condition for Equilibrium

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Concurrent Forces

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Polygon Law of Vectors

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Triangle of Forces

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion