Problem 13 - Applied Mathematics

Question

Calculate the magnitude of the resultant of two forces \(F_{1}\) and \(\mathbf{F}_{2}\). (a) \(\mathrm{F}_{1}=3 \mathrm{i}+7 \mathrm{j}\). b) \(F_{2}=\mathbf{i}-4 \mathbf{j}\) c) Both forces act at a point \(2 \mathrm{i}+\mathrm{j}\).

Step-by-Step Solution

Answer

The magnitude of the resultant force \(\textbf{F}_{R}\) is \(5\).

Step 1 - Understand the given vectors

Given two vectors are \(\textbf{F}_{1} = 3\textbf{i} + 7\textbf{j}\) and \(\textbf{F}_{2} = \textbf{i} - 4\textbf{j}\). Both act at the same point, but the position of the point does not affect the calculation of the resultant force.

Step 2 - Add the vectors

To find the resultant vector \(\textbf{F}_{R}\), sum the components of both vectors: \(\textbf{F}_{R} = (3 \textbf{i} + 7 \textbf{j}) + (\textbf{i} - 4 \textbf{j})\).

Step 3 - Combine like terms

Combine the \(\textbf{i}\) and \(\textbf{j}\) components separately: \(\textbf{F}_{R} = (3+1) \textbf{i} + (7-4) \textbf{j} = 4 \textbf{i} + 3 \textbf{j}\).

Step 4 - Calculate the magnitude of the resultant

The magnitude of \(\textbf{F}_{R}\) is calculated using the formula \(\text{Magnitude} = \sqrt{(a^2 + b^2)}\) where \(\textbf{F}_{R} = a \textbf{i} + b \textbf{j}\). Hence, \[\text{Magnitude} = \sqrt{(4^2 + 3^2)} = \sqrt{16 + 9} = \sqrt{25} = 5\].

Key Concepts

Vector Addition

Vectors are mathematical entities that have both magnitude and direction. They are often represented as arrows in a coordinate system. Adding vectors is a fundamental concept in physics and engineering.

When adding two vectors, \(\textbf{F}_{1}\) and \(\textbf{F}_{2}\), you sum their respective components. For example, if \(\textbf{F}_{1} = 3\textbf{i} + 7\textbf{j}\) and \(\textbf{F}_{2} = \textbf{i} - 4\textbf{j}\), their resultant vector \(\textbf{F}_{R}\) is found by adding the \(\textbf{i}\) and \(\textbf{j}\) components separately:

• You add the \(\textbf{i}\) components: \(3\textbf{i} + \textbf{i} = 4\textbf{i}\)


• You add the \(\textbf{j}\) components: \(7\textbf{j} - 4\textbf{j} = 3\textbf{j}\)

So, the resultant vector \(\textbf{F}_{R}\) would be \(4\textbf{i} + 3\textbf{j}\). This resultant vector represents the combined effect of both forces.

Magnitude of a Vector

The magnitude of a vector quantifies its length, regardless of its direction. To find the magnitude of a vector like \(\textbf{F}_{R} = 4\textbf{i} + 3\textbf{j}\), you use the Pythagorean theorem.

Here's the formula for the magnitude of a vector \(a\textbf{i} + b\textbf{j}\):
\[\text{Magnitude} = \sqrt{a^2 + b^2}\]

For \(\textbf{F}_{R}\) in this exercise:
\begin{align*} \text{Magnitude} & = \sqrt{(4^2 + 3^2)} \ & = \sqrt{16 + 9} \ & = \sqrt{25} \ & = 5 \end{align*}
This tells us the resultant force has a magnitude of 5 units.

Vector Components

Every vector can be broken down into its components along the coordinate axes. This makes complex vector operations simpler.

For example, the vector \(3\textbf{i} + 7\textbf{j}\) has two components:

• The \(\textbf{i}\)-component (horizontal direction) is 3


• The \(\textbf{j}\)-component (vertical direction) is 7

These components are the projections of the vector along the respective axes. To find the resultant vector \(\textbf{F}_{R}\) when combining two vectors, we sum these components separately.

This method allows us to deal with each direction independently, simplifying the calculation process and making it easier to understand the resultant vector in both magnitude and direction.