Problem 47 - Algebra and Trigonometry

Question

Resultant of Four Forces An object located at the origin in a three- dimensional coordinate system is held in equilibrium by four forces. One has magnitude 7 lb and points in the direction of the positive x-axis, so it is represented by the vector 7i. The second has magnitude 24 lb and points in the direction of the positive y-axis. The third has magnitude 25 lb and points in the direction of the negative z-axis. (a) Use the fact that the four forces are in equilibrium (that is, their sum is 0) to find the fourth force. Express it in terms of the unit vectors i, j, and k. (b) What is the magnitude of the fourth force?

Step-by-Step Solution

Answer

The fourth force is \(-7\mathbf{i} - 24\mathbf{j} + 25\mathbf{k}\) with a magnitude of \(35.36 \text{ lb}\).

Step 1: Define Known Forces as Vectors

- The first force is represented by \( \vec{F_1} = 7\mathbf{i} \). - The second force is \( \vec{F_2} = 24\mathbf{j} \).- The third force is \( \vec{F_3} = -25\mathbf{k} \).

Step 2: Set Up the Equation for Equilibrium

Since the object is in equilibrium, the sum of all forces is zero: \[ \vec{F_1} + \vec{F_2} + \vec{F_3} + \vec{F_4} = 0 \]Substituting the known forces gives us:\[ 7\mathbf{i} + 24\mathbf{j} - 25\mathbf{k} + \vec{F_4} = 0 \]

Step 3: Solve for the Unknown Force

Rearrange the equation to solve for \(\vec{F_4}\):\[ \vec{F_4} = - (7\mathbf{i} + 24\mathbf{j} - 25\mathbf{k}) \]Thus, the fourth force is:\[ \vec{F_4} = -7\mathbf{i} - 24\mathbf{j} + 25\mathbf{k} \]

Step 4: Calculate the Magnitude of the Fourth Force

The magnitude of \(\vec{F_4}\) is calculated using the formula for the magnitude of a vector:\[ |\vec{F_4}| = \sqrt{(-7)^2 + (-24)^2 + (25)^2} \]\[ |\vec{F_4}| = \sqrt{49 + 576 + 625} \]\[ |\vec{F_4}| = \sqrt{1250} \]\[ |\vec{F_4}| = 35.36 \text{ lb} \]

Key Concepts

Vector Equilibrium

When we talk about vector equilibrium, we are discussing a situation where all the forces acting on an object perfectly balance each other out. This means that the object is not accelerating in any direction, as the total sum of forces equals zero. In simpler terms, everything pushes or pulls evenly in all directions.
This concept is crucial in many areas of physics and engineering because it helps us design stable structures and machines.
In the original exercise, we have four forces playing the part to create equilibrium. The equation to express this balance is:

• Sum of all forces = 0

Forces are represented as vectors because they have both magnitude (how strong the force is) and direction. If you think of each force as created by someone playing tug-of-war, equilibrium would be the situation where no team is moving forward or backward. The forces are all tugging equally in opposite directions, balancing the system.

Magnitude of a Vector

The magnitude of a vector refers to its size or length. When describing a vector, its magnitude tells you how strong the effect of that vector is. In the context of forces, it's how powerful a force is compared to others.
To calculate a vector's magnitude, you can use the Pythagorean theorem in three dimensions. Essentially, it's like measuring the diagonal of a three-dimensional box that the vector would form with the axes. The formula is:
\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]
where \(x, y,\) and \(z\) are the components of the vector along each axis.
In our exercise, we used this formula to determine the magnitude of the fourth force:\[ |\vec{F_4}| = \sqrt{(-7)^2 + (-24)^2 + (25)^2} = 35.36 \text{ lb} \]
This tells us how hard this fourth force is "pushing" compared to the others.

Unit Vectors

Unit vectors are a simple yet fundamental concept in vector mathematics. They are vectors that have a length of exactly one unit. Think of a pointer that simply shows direction but hasn't grown in force or magnitude.
These vectors are incredibly useful because they help us express any vector's direction in space without affecting its magnitude. They are denoted as \(\mathbf{i}, \mathbf{j}, \) and \(\mathbf{k}\) for the x, y, and z axes respectively.
In the exercise, the forces are stated with respect to these unit vectors, making it easier to express their direction. For example:
\( 7\mathbf{i} \) means 7 units of force in the x-direction.
Using unit vectors simplifies calculations and represents complex ideas, like vector addition or finding resultant forces, in a clear way. They are like building blocks that help construct more detailed and varied vectors. Since they only show direction, they allow us to easily scale the vector up or down using multiplication by their magnitude. This is why unit vectors are a staple tool in physics and engineering.