Problem 54 - Algebra and Trigonometry

Question

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

The resultant force is \(\mathbf{j}\); an additional force of \(-\mathbf{j}\) is needed for equilibrium.

Step 1: Calculate the Resultant Force

To find the resultant force at point \(P\), we add the given forces \( \mathbf{F}_{1}, \mathbf{F}_{2}, \) and \( \mathbf{F}_{3} \). The sum is calculated as follows: \( \mathbf{F}_{ ext{resultant}} = \mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} = (\mathbf{i} - \mathbf{j}) + (\mathbf{i} + \mathbf{j}) + (-2\mathbf{i} + \mathbf{j}) \). By combining like terms, we get: \( \mathbf{F}_{ ext{resultant}} = 0\mathbf{i} + 1\mathbf{j} = \mathbf{j} \).

Step 2: Check for Equilibrium

A set of forces is in equilibrium if their resultant force is zero. From Step 1, the resultant force at point \(P\) is \(\mathbf{j}\). Therefore, the forces \(\mathbf{F}_{1}, \mathbf{F}_{2}, \) and \(\mathbf{F}_{3} \) are not in equilibrium as \(\mathbf{F}_{ ext{resultant}} eq 0 \).

Step 3: Determine the Additional Force

To bring the forces into equilibrium, we need to add a force that cancels out \(\mathbf{F}_{ ext{resultant}}\). The current resultant is \(\mathbf{j}\), so the additional force required is \(-\mathbf{j}\). By adding \(-\mathbf{j}\), the total resultant force will become zero, satisfying the equilibrium condition.

Key Concepts

Resultant Force

In physics, when multiple forces act on a single point, the resultant force is the vector sum of all these forces. Think of it as the 'net effect' of all the individual forces acting together. Calculating the resultant force is crucial in determining whether an object remains in equilibrium, meaning it stays at rest or moves with constant velocity.
A simple way to understand this is by adding up all the forces acting on that point. For example, if a point has forces like

• \( \mathbf{F}_1 = \mathbf{i} - \mathbf{j} \)
• \( \mathbf{F}_2 = \mathbf{i} + \mathbf{j} \)
• \( \mathbf{F}_3 = -2\mathbf{i} + \mathbf{j} \)

on it, you combine them as follows: \( \mathbf{F}_{\text{resultant}} = \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 \). By adding these forces together, you simplify terms to find \( \mathbf{F}_{\text{resultant}} = 0\mathbf{i} + 1\mathbf{j} = \mathbf{j} \).
Here, the resultant force is not zero, indicating the forces are not balanced or in equilibrium.

Vector Addition

Vector addition is essential when dealing with forces. Each force can be seen as a vector, having both magnitude and direction. The beauty of vector addition is that it accounts for both these aspects. By representing each force with its respective i (horizontal) and j (vertical) components, one can visualize and calculate how these forces work together.
Imagine adding vectors like stacking arrows, where each arrow points in a specific direction. For instance, the arrows pointing in the opposite direction cancel out each other. So, when you see \( \mathbf{F}_1 = \mathbf{i} - \mathbf{j} \) and \( \mathbf{F}_2 = \mathbf{i} + \mathbf{j} \), the i components are both \( 1\mathbf{i} \), summing up to \( 2\mathbf{i} \). But then, force \( \mathbf{F}_3 = -2\mathbf{i} + \mathbf{j} \) comes in and cancels out that \( 2\mathbf{i} \) entirely. It's these basics of vector addition that help in finding the resultant force.

Additional Force

After calculating the resultant force, we get to know if the forces are in equilibrium. If not, finding an additional force is necessary to achieve it. The idea behind the additional force is straightforward. It's the extra push or pull needed to make the total resultant force zero.
For example, if the resultant force derived from the sum of all forces acting at a point is \( \mathbf{j} \), the system is not in equilibrium. The additional force must exactly balance this, and so it should be \( -\mathbf{j} \). By adding this additional force, the new sum of forces becomes zero. This concept is pivotal in various applications, ensuring that objects either remain in a state of rest or move at a steady velocity without any acceleration.