Problem 53 - 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. $$ \begin{array}{l}{\mathbf{F}_{1}=4 \mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=3 \mathbf{i}-7 \mathbf{j}, \quad \mathbf{F}_{3}=-8 \mathbf{i}+3 \mathbf{j}} \\ {\mathbf{F}_{4}=\mathbf{i}+\mathbf{j}}\end{array} $$

Step-by-Step Solution

Answer

Resultant force is \(-4\mathbf{j}\). Additional force needed is \(4\mathbf{j}\).

Step 1: Calculate the Resultant Force

To find the resultant force, we sum up all the given forces: \( \mathbf{F}_{R} = \mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} + \mathbf{F}_{4} \). \[ \mathbf{F}_{R} = (4\mathbf{i} - \mathbf{j}) + (3\mathbf{i} - 7\mathbf{j}) + (-8\mathbf{i} + 3\mathbf{j}) + (\mathbf{i} + \mathbf{j}) \]Combine like terms:\[ \mathbf{F}_{R} = (4 + 3 - 8 + 1)\mathbf{i} + (-1 - 7 + 3 + 1)\mathbf{j} \]Simplify:\[ \mathbf{F}_{R} = 0\mathbf{i} - 4\mathbf{j} \]Thus, the resultant force is \( \mathbf{F}_{R} = -4\mathbf{j} \).

Step 2: Determine the Additional Force Required for Equilibrium

For the forces to be in equilibrium, the resultant force must be zero. Currently, \( \mathbf{F}_{R} = -4\mathbf{j} \). We need an additional force \( \mathbf{F}_{A} \) such that:\[ \mathbf{F}_{R} + \mathbf{F}_{A} = 0 \]This means:\[ -4\mathbf{j} + \mathbf{F}_{A} = 0 \]Thus, \( \mathbf{F}_{A} = 4\mathbf{j} \).The additional force required to achieve equilibrium is \( 4\mathbf{j} \).

Key Concepts

Resultant Force

The resultant force is like the sum of all individual forces acting at a specific point. Imagine you have several people pushing a sled from different directions; the resultant force tells you which way the sled will move if all these forces are combined. In our exercise, we calculated the resultant force at point \( P \) by adding up the given forces \( \mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3}, \) and \( \mathbf{F}_{4} \). By combining their components in the \( \mathbf{i} \) and \( \mathbf{j} \) directions, we found that the resultant force is \( \mathbf{F}_{R} = -4\mathbf{j} \). In simpler terms, this resultant force means there is a net force acting downwards (negative \( \mathbf{j} \)-direction) once all forces have been accounted for.

Vector Addition

Vector addition is a method used to combine multiple force vectors to find the resultant or total force. Vectors have both magnitude and direction, and just like arrows, they can be broken down into components (like \( \mathbf{i} \) for horizontal and \( \mathbf{j} \) for vertical movement).
To add vectors, align them by their components, summing up all like terms in each direction.

• Combine all \( \mathbf{i} \)-components together.
• Combine all \( \mathbf{j} \)-components together.

In our problem, we added each force's projection on the \( \mathbf{i} \) and \( \mathbf{j} \) axes and found that the resultant was \( 0\mathbf{i} - 4\mathbf{j} \). This process is visual and conceptual since it shows exactly how each force contributes to the net effect.

Force Equilibrium

Force equilibrium occurs when all forces acting on an object are perfectly balanced, resulting in no movement. At this point, the resultant force equals zero. In the exercise, we noticed the forces were not in equilibrium since the resultant was \( -4\mathbf{j} \). In practical terms, one can imagine a perfectly balanced see-saw - it won't tip over as long as weights on both sides are evenly distributed.
When forces are balanced, it means:

• The sum of all force vectors equals zero.
• There is no net motion in any direction.

Understanding equilibrium is essential in engineering, architecture, and physics since it helps in designing systems that remain stable under various conditions.

Additional Force

An additional force is a force needed to bring the system back into force equilibrium. This is often necessary when the calculated resultant force is not zero, indicating that some directional force remains. In our exercise, the existing forces left an unbalanced \( -4\mathbf{j} \) (downwards).
To counter this, we needed an additional upwards force of \( 4\mathbf{j} \) to achieve balance. By adding \( 4\mathbf{j} \) to the resultant, we bring the system to equilibrium:

• The net force becomes zero.
• The object stops moving or stays at rest.

In real life, engineers might need to calculate such additional forces to ensure structures or machines don't topple over or break under load. This ensures safety and functionality in various applications.