91Ó°ÊÓ

In physics, the resultant force is the overall force acting on an object when all the individual forces are combined. To find the resultant force, you simply add up all the forces acting on the object by summing their vector components.
Imagine you have two forces acting on a particle: Force A and Force B. The resultant force, often denoted as \(\backslashvec{F}_{R}\), can be found by adding the components of Force A and Force B.

For our exercise, Force A is given by \(\backslashvec{F}_{A} = 10 \text{N} \hat{\mathrm{i}} - 4 \text{N} \hat{\mathrm{j}}\) and Force B is \(\backslashvec{F}_{B} = 17 \text{N} \hat{\mathrm{i}} + 2 \text{N} \hat{\mathrm{j}}\). Adding these forces together, we get:

\[\backslashvec{F}_{R} = (10 + 17) \text{N} \hat{\mathrm{i}} + ((-4) + 2) \text{N} \hat{\mathrm{j}} = 27 \text{N} \hat{\mathrm{i}} - 2 \text{N} \hat{\mathrm{j}}\]
The resultant force \(\backslashvec{F}_{R}\) is a single force that represents the combined effect of both Force A and Force B.

Understanding the components of a force is crucial in solving problems involving vector quantities. Every force can be broken down into its horizontal and vertical components, typically along the x and y axes.
For instance, Force A in our exercise has components along the x-axis (\(\backslashhat{\mathrm{i}}\)) and the y-axis (\(\backslashhat{\mathrm{j}}\)). Specifically, Force A can be written as:

\(\backslashvec{F}_{A} = 10 \text{N} \hat{\mathrm{i}} - 4 \text{N} \hat{\mathrm{j}}\)

This notation means that Force A has a 10 N component along the x-axis and a -4 N component along the y-axis. Similarly, for Force B, we have:

\(\backslashvec{F}_{B} = 17 \text{N} \hat{\mathrm{i}} + 2 \text{N} \hat{\mathrm{j}}\)

• The x-component is 17 N
• The y-component is 2 N

Breaking forces into components makes it easier to perform calculations, especially when adding or subtracting forces.

Vector addition is the process of combining vectors to determine their cumulative effect. This involves adding the corresponding components of the vectors.
In our exercise, we are adding Force A and Force B to find the resultant force. Let's do a step-by-step addition:

• Add the x-components: \(10 \text{N} + 17 \text{N} = 27 \text{N}\)
• Add the y-components: \((-4 \text{N} + 2 \text{N} = -2 \text{N}\)

Therefore, the resultant force \(\backslashvec{F}_{R}\) is:

\[\backslashvec{F}_{R} = 27 \text{N} \hat{\mathrm{i}} - 2 \text{N} \hat{\mathrm{j}}\]

This resultant force combines the effects of Force A and Force B and will act in the direction specified by its components.

The direction of a force is crucial in understanding its impact on the object it acts upon. The direction is often measured relative to a defined axis, like the x-axis. To determine this direction, we typically use trigonometric functions.

For the balancing force \(\backslashvec{F}_{C}\) in our exercise, its components are \(-27 \text{N} \hat{\mathrm{i}} + 2 \text{N} \hat{\mathrm{j}}\). We can find the angle \(\backslashtheta\) relative to the x-axis using the tangent function:

\[\backslashtan(\backslashtheta) = \frac{F_{Cy}}{F_{Cx}} = \frac{2}{-27}\]
Thus, the angle is:

\[\backslashtheta = \backslashtan^{-1}\bigg(\frac{2}{-27}\bigg)\]

This calculation gives us the direction of \(\backslashvec{F}_{C}\) relative to the x-axis, which helps us understand precisely which way the force is acting.