Problem 51 - Contemporary Precalculus

Question

If forces \(\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \ldots, \boldsymbol{u}_{k}\) act on an object at the origin, the resultant force is the sum \(u_{1}+u_{2}+\cdots+u_{k} .\) The forces are said to be in equilibrium if their resultant force is \(0 .\) In Exercises 51 and \(52,\) find the resultant force and find an additional force \(v\) that, if added to the system, produces equilibrium. $$\mathbf{u}_{1}=\langle 2,5\rangle, \mathbf{u}_{2}=\langle-6,1\rangle, \mathbf{u}_{3}=\langle-4,-8\rangle$$

Step-by-Step Solution

Answer

Answer: The resultant force of the given system is \(\mathbf{u}_{R} = \langle -8, -2 \rangle\). The additional force vector needed to achieve equilibrium is \(\mathbf{v} = \langle 8, 2 \rangle\).

Step 1: Sum up the force vectors

Calculate the sum of the given force vectors by adding their respective components. Resultant force vector: \(\mathbf{u}_{R} = \mathbf{u}_{1}+\mathbf{u}_{2}+\mathbf{u}_{3} = \langle 2+(-6)+(-4), 5+1+(-8) \rangle = \langle -8, -2 \rangle\)

Step 2: Find the additional force vector to make the system in equilibrium

Calculate the additional force vector \(\mathbf{v}\) that would make the total resultant force equal to 0. To achieve equilibrium, the following equation must hold true: \(\mathbf{u}_{R} + \mathbf{v} = \mathbf{0}\) So, to find the equilibrium force vector: \(\mathbf{v} = -\mathbf{u}_{R} = \langle 8, 2 \rangle\)

Step 3: Present the resultant force and additional force vector

The resultant force is \(\mathbf{u}_{R} = \langle -8, -2 \rangle\) and the additional force vector to achieve equilibrium is \(\mathbf{v} = \langle 8, 2 \rangle\).

Key Concepts

Equilibrium of Forces

Forces are said to be in equilibrium when their combined effect results in no movement. This means that the total force acting on an object must sum up to zero. In scenarios where multiple forces are acting on an object, like in our exercise, equilibrium occurs when their resultant force is zero.

This can be expressed mathematically as the sum of all the forces \( \mathbf{u}_{1} + \mathbf{u}_{2} + \mathbf{u}_{3} + \ldots + \mathbf{u}_{k} = \mathbf{0} \). If the forces add up to a zero vector, it means they perfectly balance each other out, so no net force acts on the object.

• When forces are in equilibrium, the object remains stationary or moves with a constant velocity.
• Finding equilibrium often involves calculating an additional vector that offsets the current resultant force, making the net effect zero.

The additional force in our exercise, denoted as \(\mathbf{v}\), is specifically calculated to neutralize the current resultant force.

Resultant Force

The resultant force is a single vector that is created by summing up all individual force vectors acting on an object. You can think of it as the overall effect of multiple forces combined. In our exercise, the forces \(\mathbf{u}_{1} = \langle 2,5\rangle\), \(\mathbf{u}_{2} = \langle-6,1\rangle\), and \(\mathbf{u}_{3} = \langle-4,-8\rangle\) are summed to find the resultant force.

To get the resultant force, you simply add the corresponding components of each vector. For example, the resulting vector for our set of forces is calculated as:\[ \mathbf{u}_{R} = \langle 2 + (-6) + (-4), 5 + 1 + (-8) \rangle = \langle -8, -2 \rangle \]

• The resultant force gives a quick summary of the total impact of all forces working together.
• It shows both the magnitude and direction of the overall force.
• This vector is key to determining whether further forces are needed to achieve equilibrium.

By assessing the resultant force, we can determine needed adjustments to bring a system of forces into equilibrium.

Vector Components

Understanding vector components is vital for adding vectors. Each vector has two components: in this case, let's consider the horizontal and vertical components, often represented as \(x\) and \(y\), respectively.

In our exercise, each force \(\mathbf{u}_{i}\) is given as a vector with two components. These components are added separately to find the resultant force. Here's how it works:

• Horizontally (\(x\)-component): Add the first number from each vector. For example, \(2 + (-6) + (-4)\) yields \(-8\).
• Vertically (\(y\)-component): Add the second number from each vector. For example, \(5 + 1 + (-8)\) results in \(-2\).

Together, these components form a new vector: \(\langle -8, -2 \rangle\). This process is essential because it allows for precise control over how different forces interact in different directions, which ultimately affects the final resultant vector and any adjustments needed for equilibrium. Each vector's components determine not just the force's size but also its orientation.