Problem 52 - 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. $$ \mathbf{F}_{1}=\langle 3,-7\rangle, \quad \mathbf{F}_{2}=\langle 4,-2\rangle, \quad \mathbf{F}_{3}=\langle- 7,9\rangle $$

Step-by-Step Solution

Answer

The resultant force is \( \langle 0, 0 \rangle \), so the forces are in equilibrium and no additional force is needed.

Step 1: Calculate the Resultant Force

To find the resultant force, sum all the given force vectors. The resultant force \( \mathbf{F}_{R} \) is given by: \[\mathbf{F}_{R} = \mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3}\] Substitute the force vectors: \[\mathbf{F}_{R} = \langle 3, -7 \rangle + \langle 4, -2 \rangle + \langle -7, 9 \rangle\] Calculate the sum component-wise: \[x\text{-component: } 3 + 4 - 7 = 0\] \[y\text{-component: } -7 - 2 + 9 = 0\] Thus, the resultant force is \( \mathbf{F}_{R} = \langle 0, 0 \rangle \).

Step 2: Determine if the Forces are in Equilibrium

Since the resultant force \( \mathbf{F}_{R} = \langle 0, 0 \rangle \), this means that the net force at point \( P \) is zero. Thus, the forces are already in equilibrium.

Step 3: Checking for Additional Force Requirement

Since the forces are already in equilibrium (\( \mathbf{F}_{R} = \langle 0, 0 \rangle \)), no additional force is required to achieve equilibrium.

Key Concepts

Resultant Force

The concept of a resultant force is central when dealing with multiple forces acting at a single point. To determine the resultant force, you must add together all the individual force vectors that are applied at that point. The resultant force is essentially the single force that would have the same effect as the combined action of the original forces.
In mathematical terms, the sum of the forces can be written as:

• \( \mathbf{F}_{R} = \mathbf{F}_{1} + \mathbf{F}_{2} + \cdots + \mathbf{F}_{n} \)

This formula sums each component of the vectors separately, meaning all x-components are added together and all y-components are similarly summed. A practical example of this is when forces \( \mathbf{F}_{1} = \langle 3, -7 \rangle \), \( \mathbf{F}_{2} = \langle 4, -2 \rangle \), and \( \mathbf{F}_{3} = \langle -7, 9 \rangle \) result in a resultant force of \( \mathbf{F}_{R} = \langle 0, 0 \rangle \), indicating that the forces balance one another out at point \( P \).
Understanding how to calculate the resultant force is critical in engineering and physics, as it dictates how structures will respond to various loads and stresses.

Vector Addition

Vector addition is a fundamental concept used in physics and engineering for combining multiple vectors. It's a way to calculate a single vector, the resultant, that has the same effect as applying all the separate vectors together.
To perform vector addition, follow these simple steps:

• Align vectors tail to head in sequence.
• Add corresponding components: the sum of all x-components and y-components.
• The resultant vector extends from the tail of the first vector to the head of the last vector in the series.

For instance, with vector \( \mathbf{A} = \langle 3, 4 \rangle \) and \( \mathbf{B} = \langle -1, 2 \rangle \), adding these together results in a new vector \( \mathbf{C} = \langle 2, 6 \rangle \).
This means the impacts of moving in the direction of \( \mathbf{A} \) followed by \( \mathbf{B} \) combine into a single vector, shown by \( \mathbf{C} \).This idea not only makes calculations easier but also helps predict outcomes in real-world scenarios like determining displacement, force, or velocity.

Equilibrium Condition

The equilibrium condition is vital in understanding how different forces balance each other. A system is in equilibrium when the net force acting on it is zero, meaning there is no unbalanced force causing acceleration. This state can be described mathematically using the concept of a resultant force.
A key aspect to note is that the equilibrium condition ensures stability. For forces \( \mathbf{F}_{1}, \mathbf{F}_{2}, \mathbf{F}_{3} \) acting on a point, if their sum \( \mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} = \mathbf{0} \), the point is in equilibrium, indicating no net movement.

• Static Equilibrium: The object is stationary, with no net force or torque.
• Dynamic Equilibrium: The object moves with constant velocity without acceleration.

When forces are in equilibrium, such as with our example vectors resulting in \( \mathbf{F}_{R} = \langle 0, 0 \rangle \), no additional force is needed to maintain this balance.
Recognizing when equilibrium is present helps in designing stable systems or analyzing the forces at play in a balanced object.