Problem 4 - Applied Mathematics

Question

Forces represented by \(3 \mathbf{i}+5 \mathbf{j}, \mathbf{i}-2 \mathbf{j}\) and \(-3 \mathbf{i}+\mathbf{j}\) together with a fourth force F act on a particle. If the resultant force is represented by \(4 \mathrm{i}+\mathrm{j}\), find \(\mathbf{F}\).

Step-by-Step Solution

Answer

\( \textbf{F} = 3 \textbf{i} - 3 \textbf{j} \)

Step 1 - Express the Resultant Force

The resultant force is given as: \(4 \textbf{i} + \textbf{j}\)

Step 2 - Sum the Given Forces

Add the three given forces: \((3 \textbf{i} + 5 \textbf{j}) + (\textbf{i} - 2 \textbf{j}) + (-3 \textbf{i} + \textbf{j}) = (3 + 1 - 3) \textbf{i} + (5 - 2 + 1) \textbf{j} = \textbf{i} + 4 \textbf{j}\)

Step 3 - Introduce the Unknown Force \(\textbf{F}\)

Let \(\textbf{F} = a \textbf{i} + b \textbf{j}\). Thus, \(\textbf{F} + \textbf{i} + 4 \textbf{j} = 4 \textbf{i} + \textbf{j}\)

Step 4 - Solve for Components of \(\textbf{F}\)

Set up the equations by comparing components: \(a + 1 = 4\) and \(b + 4 = 1\). Solve for \(a\) and \(b\): \(a = 3\), and \(b = -3\)

Step 5 - Write the Force \(\textbf{F}\)

Combine the components to form \(\textbf{F}\): \(\textbf{F} = 3 \textbf{i} - 3 \textbf{j}\)

Key Concepts

Vector Addition

Vector addition is a way to combine multiple vectors to produce a single resultant vector. This involves adding the corresponding components of the vectors. When you are given vectors in the form of \(\textbf{i}\) and \(\textbf{j}\) units, you simply add the coefficients of \(\textbf{i}\) together and those of \(\textbf{j}\) together. For instance, if we have three vectors \((3 \textbf{i} + 5 \textbf{j}) + (\textbf{i} - 2 \textbf{j}) + (-3 \textbf{i} + \textbf{j})\), we can find the resultant by summing each component:

\((3 + 1 - 3) \textbf{i}\) and \((5 - 2 + 1) \textbf{j})\).
This yields \(\textbf{i} + 4 \textbf{j}\).

Vector addition is crucial when dealing with forces, as forces are vector quantities; they have both a magnitude and a direction. This method allows us to understand the cumulative effect of multiple forces acting on a body.

Equilibrium of Forces

Equilibrium of forces is a condition where the sum of all forces acting on a particle is zero, meaning there is no net force. For a particle to be in equilibrium, the vector sum of all the forces must equal the zero vector. However, in our problem, we are given that the resultant force is \( 4 \textbf{i} + \textbf{j} \). This means the combined effect of the given forces and the unknown force \( \textbf{F} \) should be equal to \( 4 \textbf{i} + \textbf{j} \).

The task involves ensuring that the resultant force you calculate matches this condition exactly. By summing the given forces and then adjusting with an unknown force \( \textbf{F} = a \textbf{i} + b \textbf{j} \), you can solve to find the exact components needed for \(\textbf{F}\) to achieve equilibrium. You will compare and solve the components individually:

• For the \(\textbf{i}\) component: \(a + 1 = 4\)
• For the \(\textbf{j}\) component: \(b + 4 = 1\)

This results in \(a = 3\) and \(b = -3,\) giving us \( \textbf{F} = 3 \textbf{i} - 3 \textbf{j} \).

Vector Components

Understanding vector components is key to solving problems involving forces. A vector can be broken down into its individual components along the coordinate axes. For 2D vectors, these components are typically aligned with the \(\textbf{i}\) and \(\textbf{j}\) directions, which correspond to the x and y axes respectively. For example, the force \(3 \textbf{i} + 5 \textbf{j} \) means it has a horizontal component of \(3 \) and a vertical component of \(5 \).

This decomposition is useful because it lets you treat each direction independently when performing vector addition or when finding the resultant force. In our exercise:

• The combined force: \( \textbf{i} + 4 \textbf{j} \)
• The resultant force: \( 4 \textbf{i} + \textbf{j} \)

To match these, we introduce a new vector \( \textbf{F} = 3 \textbf{i} - 3 \textbf{j} \), which compensates for the difference in each direction, ensuring our final sum equates to the given resultant force. Components simplify these calculations by isolating each directional element.