Problem 68 - 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. $$ \mathbf{F}_{1}=\langle 3,-7\rangle, \quad \mathbf{F}_{2}=\langle 4,-2\rangle, \quad \mathbf{F}_{3}=\langle- 7,9\rangle $$

Step-by-Step Solution

Answer

The resultant force is \( \langle 0, 0 \rangle \), and no additional force is needed for equilibrium.

Step 1: Understand the Resultant Force

To find the resultant force acting at point \( P \), sum the vectors \( \mathbf{F}_{1}, \mathbf{F}_{2}, \) and \( \mathbf{F}_{3} \). The resultant force is calculated by adding the corresponding components of the vectors.

Step 2: Calculate the Resultant Force

Sum the components of the vectors:For the \( x \)-components: \( 3 + 4 - 7 = 0 \)For the \( y \)-components: \( -7 - 2 + 9 = 0 \)Thus, the resultant force \( \mathbf{R} = \langle 0, 0 \rangle \).

Step 3: Determine Equilibrium

Since the resultant force \( \mathbf{R} \) is \( \langle 0, 0 \rangle \), the forces are already in equilibrium. No additional force is required.

Key Concepts

Understanding Resultant Force

A resultant force is essentially the sum of all the forces acting at a point. Picture a game of tug-of-war; if the forces acting in all directions result in no movement, the forces are in balance, meaning their resultant force is zero. When discussing forces at a point, we say they are in equilibrium when their resultant is zero.
The key to finding this resultant force is adding individual forces together. This involves adding up their directional components, such as their horizontal and vertical slices in a straightforward math operation. If the sum results in a zero vector, the forces don't cause any change, keeping the system in equilibrium.
This brings us to the notion that when forces are in equilibrium, there's no need for any additional balancing force because the existing forces neutralize each other perfectly.

Basics of Vectors

Vectors are fundamental building blocks in physics and mathematics, used to represent quantities that have both magnitude and direction. Imagine arrows pointing from one point to another; these arrows show how much and in which direction a force acts. Each vector is expressed by its components, typically along the x and y axes.
For example, the vector \( \mathbf{F}_1 = \langle 3, -7 \rangle \) means it has a magnitude of 3 units in the x-direction and -7 units in the y-direction. It's crucial to understand that vectors help simplify complex motion by breaking it into manageable, directional pieces. This abstraction allows us to apply simple mathematical rules to solve problems involving forces, motion, and more.
By visualizing and understanding vectors, we gain a clearer insight into how different forces interact and combine, which leads us to the concept of vector addition.

The Art of Vector Addition

Vector addition is like finding a single vector that combines the influences of multiple vectors. It's akin to mixing colors to get a new shade. In physics, this new vector represents collectively where all the forces want to push or pull. To add vectors, we simply add their corresponding components.
Consider the problem at hand where we have vectors \( \mathbf{F}_1 = \langle 3, -7 \rangle \), \( \mathbf{F}_2 = \langle 4, -2 \rangle \), and \( \mathbf{F}_3 = \langle -7, 9 \rangle \). For the x-components: add 3 (from \( \mathbf{F}_1 \)) to 4 (from \( \mathbf{F}_2 \)) and -7 (from \( \mathbf{F}_3 \)), resulting in 0. For the y-components: sum up -7, -2, and 9, which also yields 0.
This process shows that the vector sum, also called the resultant, is zero. This zero vector indicates no net push or pull, confirming equilibrium. Practicing vector addition helps us precisely determine how different forces interact and neutralize or reinforce each other.