Problem 69 - 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. $$ \begin{array}{l}{\mathbf{F}_{1}=4 \mathbf{i}-\mathbf{j}, \quad \mathbf{F}_{2}=3 \mathbf{i}-7 \mathbf{j}, \quad \mathbf{F}_{3}=-8 \mathbf{i}+3 \mathbf{j}} \\ {\mathbf{F}_{4}=\mathbf{i}+\mathbf{j}}\end{array} $$

Step-by-Step Solution

Answer

Resultant force: \(0\mathbf{i} - 4\mathbf{j}\) and additional force needed: \(0\mathbf{i} + 4\mathbf{j}\) for equilibrium.

Step 1: Calculate the Sum of the Forces

To find the resultant force, we must sum all given forces. Start by adding the components of the vectors separately for the \( \mathbf{i} \) and \( \mathbf{j} \) components.\[\text{Net } \mathbf{i} \text{ component} = (4 + 3 - 8 + 1) = 0\]\[\text{Net } \mathbf{j} \text{ component} = (-1 - 7 + 3 + 1) = -4\]The resultant force \( \mathbf{F}_{\text{resultant}} \) at point \( P \) is then:\[\mathbf{F}_{\text{resultant}} = 0 \mathbf{i} - 4 \mathbf{j}\]

Step 2: Evaluate Equilibrium Condition

For the forces to be in equilibrium, the resultant force must be zero. Here, the component along \( \mathbf{i} \) is zero, but the component along \( \mathbf{j} \) is \(-4\). This means the forces are not currently in equilibrium, as the resultant force is not zero.

Step 3: Determine the Additional Force Required for Equilibrium

To achieve equilibrium, we need to add a force that will cancel out the \(-4\mathbf{j}\) component of the resultant force. Therefore, the additional force needed is:\[\mathbf{F}_{\text{additional}} = 0 \mathbf{i} + 4 \mathbf{j}\]This will ensure that the total force at point \( P \) is zero, satisfying the equilibrium condition.

Key Concepts

Resultant Force

In physics, especially when dealing with forces, it is crucial to understand the concept of Resultant Force. The resultant force is essentially the single force that represents the vector sum of all individual forces acting on a point. To determine if multiple forces acting on a point are balanced, we calculate the resultant force by adding all the vector components.
For example, given four vectors with components along directions labeled as \( \mathbf{i} \) and \( \mathbf{j} \), the resultant force would be calculated by summing up the components in each direction separately. If after performing this addition, the resultant force is a zero vector, it would mean the forces are perfectly cancelling each other out. However, if there is any component left over, like a non-zero \( \mathbf{j} \) component, for instance, the sum of the forces does not lead to balance or equilibrium.

• Resultant Force reflects the cumulative effect of all forces acting on a point.
• Calculated by performing vector addition on all force vectors involved.
• If the resultant force is zero, forces are balanced; otherwise, they are not.

Vector Addition

Vector Addition is a mathematical operation that is used extensively in physics to combine forces. When forces are represented as vectors, each having both a magnitude and direction, vector addition allows us to find how these forces interact collectively.
In terms of force components, this addition means adding the corresponding \( \mathbf{i} \) components and the corresponding \( \mathbf{j} \) components separately. In our exercise scenario, you'd take each force vector and determine its contribution to each directional component, then sum up all the contributions.
For instance, the total \( \mathbf{i} \) component is computed by adding all the \( \mathbf{i} \) parts of the vectors. Likewise, the \( \mathbf{j} \) components are summed to find the total effect along that direction. Through this process, vector addition helps us solve problems involving multiple forces on a single point.

• Vectors are combined by adding corresponding components: \( \mathbf{i} \) with \( \mathbf{i} \), \( \mathbf{j} \) with \( \mathbf{j} \).
• Enables us to find the effective force or resultant on a point.
• Crucial for understanding how forces contribute to motion or equilibrium.

Equilibrium Condition

To achieve balance, or equilibrium, the forces acting at a point must satisfy the Equilibrium Condition. This core principle states that the sum of all forces acting on the point must be zero.
Equilibrium Condition ensures that all the forces balance out, with no net force leading to movement. When forces are added vectorially and the resultant force is zero, the point remains at rest or moves uniformly without any acceleration. However, as illustrated in the exercise, if the resultant force isn't zero, an additional force may be needed to achieve equilibrium.
In practical terms, verifying equilibrium involves checking that both horizontal and vertical components (\( \mathbf{i} \) and \( \mathbf{j} \)) together result in no net effect. If any component isn't zero, an additional force in the opposite direction might be required to nullify it, achieving equilibrium.

• Equilibrium is achieved when resultant force is zero.
• Ensures that there is no net force causing acceleration.
• Sometimes requires an additional force to cancel any non-zero components of resultant force.