Problem 43 - Trigonometry

Question

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Resultant Force A heavy log is dragged across the ground by two horses pulling on ropes (Figure 9). The magnitudes of the tension forces in the direction of the ropes are 58 pounds and 73 pounds. If the angle between the ropes is \(26^{\circ}\), find the magnitude of the resultant force.

Step-by-Step Solution

Answer

The magnitude of the resultant force is approximately 127.68 pounds.

Step 1: Understand the Problem and Setup

We need to find the resultant force of two forces acting at an angle. A resultant force is the vector sum of two or more forces. Here, we have two tension forces with magnitudes of 58 pounds and 73 pounds, having an angle of \(26^\circ\) between them.

Step 2: Apply the Law of Cosines

To find the magnitude of the resultant force \( R \), apply the law of cosines. The formula is:\[R = \sqrt{A^2 + B^2 + 2AB \cdot \cos(\theta)}\]where \( A = 58 \) pounds, \( B = 73 \) pounds, and \( \theta = 26^\circ \).

Step 3: Compute Cosine of the Angle

Calculate the cosine of the angle \( \theta = 26^\circ \) using a calculator. \[ \cos(26^\circ) \approx 0.8988 \]

Step 4: Substitute Values into the Equation

Substitute the known values into the law of cosines equation:\[R = \sqrt{58^2 + 73^2 + 2 \times 58 \times 73 \times 0.8988}\]

Step 5: Simplify the Expression

Calculate each part of the equation:\[58^2 = 3364, \quad 73^2 = 5329, \quad 2 \times 58 \times 73 \times 0.8988 = 7614.57\]Sum these values:\[R = \sqrt{3364 + 5329 + 7614.57} = \sqrt{16307.57}\]

Step 6: Calculate the Resultant Force

Find the square root of the sum:\[R \approx 127.68 \text{ pounds}\]

Step 7: Conclude the Result

The magnitude of the resultant force acting on the log is approximately 127.68 pounds.

Key Concepts

Resultant Force

In physics, the concept of a resultant force is crucial when dealing with multiple forces acting on an object. It essentially represents the single force which has the same effect on the object as all the original forces combined.

Calculating the resultant force requires vector addition because forces have both magnitude and direction. When two forces act at an angle to one another, they don't just add up like simple numbers. Their directions matter too.

• The problem involves two forces: 58 pounds and 73 pounds.
• The angle between these forces is 26°.
• We need to find the single force that would have the same effect as the two forces combined.

This problem is solved using the Law of Cosines. This helps determine the magnitude of the resultant force when two sides and the included angle are known.

Law of Cosines

The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of an included angle. It is particularly useful in situations where two forces are acting at an angle to one another. Here's the general formula:

• \[c^2 = a^2 + b^2 - 2ab imes ext{cos}(C)\]

In the context of force vectors, we adapt this formula to find the magnitude of the resultant force. If we denote the magnitudes of the forces by \(A\) and \(B\), and the angle between them by \(\theta\), the formula becomes:

• \[R = \sqrt{A^2 + B^2 + 2AB \cdot \text{cos}(\theta)}\]

It's important to note the differences:

• Instead of subtracting the middle term, we add it because forces are combined.
• The cosine of the angle accounts for the direction and orientation of the forces.

By using the Law of Cosines, we ensure that our calculations take both the magnitudes and the angle into consideration.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a single vector, known as the resultant vector, which represents the cumulative effect. Each vector has a magnitude and a direction — akin to forces having strength and direction.

When two vectors are not aligned along the same line, simple arithmetic addition is not possible. Thus, when forces of 58 pounds and 73 pounds act at an angle of 26 degrees, they need to be added using vector addition methods.

• The simplest case is when vectors are along the same line—just add their magnitudes considering their directions.
• Here, because of the angle, we must consider trigonometric relationships.
• The resultant is found using the vector addition formula based on the law of cosines, which brings into account both magnitude and direction differences.

Vector addition thus provides a robust framework to combine forces realistically accounting for both magnitude and angle.

Bearing and Heading

When navigating or interpreting forces in real-world scenarios, the concepts of bearing and heading are vital. Bearing specifically relates to the direction from one point to another and is often measured in degrees, starting from due north and moving clockwise.

In the context of this exercise, the direction of the forces (bearing) is crucial to calculate the magnitude and effect of the resultant force.

• Bearing is often used in navigation, aviation, and sea travel.
• It provides a standard method for describing direction in a precise manner, relying on fixed points such as North (0°), East (90°), South (180°), and West (270°).

Understanding bearings fully enhances one’s ability to engage with navigation problems and solve them using trigonometry effectively. Even in scenarios like this, where forces rather than directions are discussed, the same foundational understanding applies.