Problem 70 - Algebra and Trigonometry

Question

\(67-72\) . Equilibrium of Forces The forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \ldots, \mathbf{F}_{n}\) acting at the same point \(P\) are said to be in equilibrium if the resultant force is zero, that is, if \(\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n}=0 .\) Find (a) the resultant forces acting at \(P,\) and (b) the additional force required (if any) for the forces to be in equilibrium. $$ \mathbf{F}_{1}=\mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=\mathbf{i}+\mathbf{j}, \quad \mathbf{F}_{3}=-2 \mathbf{i}+\mathbf{j} $$

Step-by-Step Solution

Answer

Resultant force is \( \mathbf{j} \); Additional force needed: \(-\mathbf{j} \).

Step 1: Understand the Problem

We need to determine the resultant force of the given forces \( \mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3} \) and check if any additional force is required for them to be in equilibrium at point \( P \). Equilibrium means the sum of all forces must equal zero.

Step 2: Calculate Resultant Force

Calculate the resultant force by adding the vectors \( \mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3} \). \[\mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} = (\mathbf{i} - \mathbf{j}) + (\mathbf{i} + \mathbf{j}) + (-2 \mathbf{i} + \mathbf{j})\]Combine like terms:\[(1+1-2)\mathbf{i} + (-1+1+1)\mathbf{j} = 0\mathbf{i} + \mathbf{j}\]Thus, the resultant force \( \mathbf{R} = \mathbf{j} \).

Step 3: Determine Additional Force for Equilibrium

For equilibrium, the resultant force must be zero. We found \( \mathbf{R} = \mathbf{j} \). Therefore, the additional force needed to achieve equilibrium is \(-\mathbf{j}\) to cancel out \( \mathbf{j} \). Hence, the additional force \( \mathbf{F}_{4} = -\mathbf{j} \).

Key Concepts

Resultant Force

Understanding the concept of resultant force is crucial when examining multiple forces acting at a single point. A resultant force is essentially the single force that has the same effect as all the individual forces acting together. When forces are acting on a point, they can either balance each other out or create a net force, which is the resultant. To find the resultant force, we need to sum all the individual force vectors.

In our example, we have three forces:

• \(\mathbf{F}_{1} = \mathbf{i} - \mathbf{j}\)
• \(\mathbf{F}_{2} = \mathbf{i} + \mathbf{j}\)
• \(\mathbf{F}_{3} = -2\mathbf{i} + \mathbf{j}\)

When we add them, the calculations show:\[\mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} = 0\mathbf{i} + \mathbf{j} \]This gives us a resultant force of \(\mathbf{j}\), meaning there is still a net force acting in the positive y-direction, which prevents equilibrium.

Vector Addition

Vector addition is a method used to find a resultant vector by adding several vectors together geometrically or algebraically. Each vector has both a magnitude and a direction, typically expressed in terms of unit vectors, such as \(\mathbf{i}\) and \(\mathbf{j}\) for horizontal and vertical components respectively. This makes vector addition essential in physics, especially in determining how different forces interact.

The process involves breaking down vectors into their components. In our case, we're given three vectors, and we look at how each contributes horizontally and vertically. Here, we add up the \(\mathbf{i}\)-components:\[1 + 1 - 2 = 0\]and the \(\mathbf{j}\)-components:\[-1 + 1 + 1 = 1\]Thus, by adding their respective components, we conclude that the sum results in a net vector \(0\mathbf{i} + \mathbf{j}\), indicating these vectors are not balanced since the resultant is not zero.

Force Equilibrium

Force equilibrium occurs when all the forces at a point effectively cancel each other out, leading to a situation where there is no net force acting. This means that the resultant force vector must be zero \((0,0)\), ensuring that the system is stable or in a state of rest. To achieve equilibrium, any extra force not neutralized by others must be addressed by introducing an additional counter force.

For equilibrium, our forces must add to zero. Here, the resultant \(\mathbf{j}\) indicates a lack of equilibrium because an upward force still exists. To resolve this and achieve equilibrium, we introduce an additional force \[\mathbf{F}_{4} = -\mathbf{j}\]This force, when added to our previous resultant, precisely cancels \(\mathbf{j}\), thus finally balancing the forces:\[\mathbf{j} + (-\mathbf{j}) = 0\]This ensures no net force is acting on point \(P\), satisfying the condition for equilibrium.