Problem 67 - Algebra and Trigonometry

Question

\(67-72\) . 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 \( \langle 5, -3 \rangle \). The additional force required for equilibrium is \( \langle -5, 3 \rangle \).

Step 1: Sum the forces

To find the resultant force at point \( P \), we sum the component vectors of the given forces. For the forces \( \mathbf{F}_{1} = \langle 2, 5 \rangle \) and \( \mathbf{F}_{2} = \langle 3, -8 \rangle \), the resultant force \( \mathbf{R} \) is: \[\mathbf{R} = \mathbf{F}_{1} + \mathbf{F}_{2} = \langle 2 + 3, 5 - 8 \rangle = \langle 5, -3 \rangle\]Thus, the resultant force \( \mathbf{R} = \langle 5, -3 \rangle \).

Step 2: Check for equilibrium

A system of forces is in equilibrium if the resultant force is zero. We already calculated the resultant force as \( \langle 5, -3 \rangle \). Since this is not the zero vector \( \langle 0, 0 \rangle \), the forces are not in equilibrium yet.

Step 3: Calculate additional force needed for equilibrium

To bring the forces into equilibrium, an additional force \( \mathbf{F}_{3} \) must cancel out the resultant force. We need \(\mathbf{R} + \mathbf{F}_{3} = \langle 0, 0 \rangle\). This means \(\mathbf{F}_{3} = -\mathbf{R} = -\langle 5, -3 \rangle = \langle -5, 3 \rangle\). Hence, the additional force required to achieve equilibrium is \( \langle -5, 3 \rangle \).

Key Concepts

Resultant Force

When discussing forces acting at the same point, the idea of a **resultant force** comes into play. The resultant force is essentially a single force that represents the combined effect of two or more forces acting at that point. Imagine multiple ropes pulling from different directions; the resultant force is like a single rope that would have the same effect as all the rest combined.

To find the resultant force, one adds up all the individual force vectors. In mathematical terms, if we have forces \( \mathbf{F}_1, \mathbf{F}_2, \ldots, \mathbf{F}_n \), the resultant force \( \mathbf{R} \) is given by:

• \( \mathbf{R} = \mathbf{F}_1 + \mathbf{F}_2 + \cdots + \mathbf{F}_n \)

This sum uses vector addition, taking into account both the magnitude and direction of each force. As demonstrated in the solution, for forces \( \mathbf{F}_1 = \langle 2, 5 \rangle \) and \( \mathbf{F}_2 = \langle 3, -8 \rangle \), their resultant force at point \( P \) is calculated as:

• \( \mathbf{R} = \langle 5, -3 \rangle \)

This outcome indicates the overall direction and strength of the combined forces.

Vector Addition

In physics and mathematics, the process of **vector addition** is crucial when dealing with forces. Vectors are quantities that have both magnitude and direction, and forces are classic examples of vectors. When adding vectors, each component (usually in two or three dimensions) should be added separately.

For example, consider two vectors \( \mathbf{F}_1 = \langle 2, 5 \rangle \) and \( \mathbf{F}_2 = \langle 3, -8 \rangle \). To determine their sum:

• Add their corresponding components.
• The first component from each vector is combined: \( 2 + 3 = 5 \).
• The second components are combined: \( 5 - 8 = -3 \).

Thus, the sum of these vectors is \( \langle 5, -3 \rangle \). Vector addition is not just simple arithmetic; it requires consideration of both direction and magnitude for accurate results.

This practice is fundamental in determining the resultant force, which is essential for the analysis of physical situations involving multiple forces acting at a point.

Force Equilibrium

**Force equilibrium** is a key concept when analyzing systems where multiple forces are involved. A system is in equilibrium when all forces balance each other out perfectly, resulting in no net force acting on the system. For a set of forces to be in equilibrium, their vector sum (or resultant force) must be zero.

In mathematical terms, if forces \( \mathbf{F}_1, \mathbf{F}_2, \ldots, \mathbf{F}_n \) are in equilibrium, then:

• \( \mathbf{F}_1 + \mathbf{F}_2 + \cdots + \mathbf{F}_n = \langle 0, 0 \rangle \)

In the example provided, the resultant force \( \langle 5, -3 \rangle \) implies that the system is not in equilibrium, because it is not the zero vector \( \langle 0, 0 \rangle \).

To achieve equilibrium, an additional force that exactly opposes the resultant (\( \mathbf{F}_3 = \langle -5, 3 \rangle \)) must be applied. This additional force cancels out the effects of the initial forces, bringing the system to a state of balance where no motion occurs unless further forces are applied.