Problem 51 - Algebra and Trigonometry

Question

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. $$ \mathbf{F}_{1}=\langle 2,5\rangle, \quad \mathbf{F}_{2}=\langle 3,-8\rangle $$

Step-by-Step Solution

Answer

The resultant force is \( \langle 5, -3 \rangle \). An additional force \( \langle -5, 3 \rangle \) is required for equilibrium.

Step 1: Calculate Resultant Force

To find the resultant force, we need to add the vectors \( \mathbf{F}_{1} \) and \( \mathbf{F}_{2} \). Each vector is defined by its components, so we add the corresponding components together: \[ \mathbf{F}_{R} = \mathbf{F}_{1} + \mathbf{F}_{2} = \langle 2, 5 \rangle + \langle 3, -8 \rangle = \langle 2+3, 5-8 \rangle = \langle 5, -3 \rangle \]. Therefore, the resultant force at point \( P \) is \( \langle 5, -3 \rangle \).

Step 2: Check for Equilibrium

A system of forces is in equilibrium if the resultant force is zero, meaning both components of the resultant vector must be zero. The current resultant force is \( \langle 5, -3 \rangle \). Since both components are not zero, the forces are not in equilibrium.

Step 3: Determine Additional Force for Equilibrium

To achieve equilibrium, we need an additional force \( \mathbf{F}_{A} = \langle x, y \rangle \) such that the total resultant becomes zero. Set up the equation: \( \langle 5, -3 \rangle + \langle x, y \rangle = \langle 0, 0 \rangle \). Solving for \( x \) and \( y \), we get \( x = -5 \) and \( y = 3 \). Thus, the additional force required is \( \mathbf{F}_{A} = \langle -5, 3 \rangle \).

Key Concepts

Resultant Force

The concept of the "resultant force" is essential when analyzing systems with multiple forces. It represents the single force that can replace all the individual forces acting on a point, without changing the physical scenario.

In the context of forces, like those in our problem with vectors \( \mathbf{F}_{1} \) and \( \mathbf{F}_{2} \), we're looking to combine these forces into one. This is done by performing vector addition. The resultant force \( \mathbf{F}_{R} \) is found by simply adding the corresponding components of the vectors together:

• For the x-component: add 2 from \( \mathbf{F}_{1} \) to 3 from \( \mathbf{F}_{2} \) to get 5.
• For the y-component: add 5 from \( \mathbf{F}_{1} \) to -8 from \( \mathbf{F}_{2} \) to get -3.

Thus, the resultant force is \( \langle 5, -3 \rangle \). This force provides a comprehensive picture of the effect of all the forces acting at point \( P \). Without this resultant force being zero, the system is not in equilibrium.

Vector Addition

Understanding vector addition is vital when working with multiple forces. Vectors have both magnitude and direction, which means they're not added like regular numbers, but via their components.

For our example, the two vectors \( \mathbf{F}_{1} = \langle 2,5 \rangle \) and \( \mathbf{F}_{2} = \langle 3,-8 \rangle \) are added component-wise:

• The x-components: 2 (from \( \mathbf{F}_{1} \)) plus 3 (from \( \mathbf{F}_{2} \)) results in 5.
• The y-components: 5 (from \( \mathbf{F}_{1} \)) plus -8 (from \( \mathbf{F}_{2} \)) results in -3.

This geometrical addition results in a new vector \( \langle 5, -3 \rangle \) that shows the overall impact of both forces in terms of direction and strength.

Vector addition provides clarity on how forces interact, ensuring a clear understanding of how they combine to produce a new, singular effect.

Force Components

The "force components" of a vector are its individual horizontal and vertical parts. These components help us understand how a force acts in different directions. Each vector can split into these parts: an x-component representing horizontal influence and a y-component for vertical influence.

In the exercise, vector \( \mathbf{F}_{1} = \langle 2, 5 \rangle \) has:

• An x-component of 2
• A y-component of 5

Similarly, vector \( \mathbf{F}_{2} = \langle 3, -8 \rangle \) has:

• An x-component of 3
• A y-component of -8

Adding these components helps evaluate the final movement in both x and y directions, contributing to the calculation of the resultant force.

This component approach is essential for visualizing, analyzing, and predicting the behavior of objects under the influence of multiple forces, as seen in complex physics problems.