Problem 21 - Algebra 2 and Trigonometry

Question

When two forces act on an object, the resultant force is the single force that would have produced the same result. When the magnitudes of the two forces are represented by the lengths of two sides of a parallelogram, the resultant can be represented by the length of the diagonal of the parallelogram. If forces of 12 pounds and 18 pounds act at an angle of \(75^{\circ}\) , what is the magnitude of the resultant force to the nearest hundredth pound?

Step-by-Step Solution

Answer

The magnitude of the resultant force is approximately 24.07 pounds.

Step 1: Understand the Concept of Resultant Force

When two forces are acting at an angle to each other, their effect can be represented as a parallelogram. We can calculate the resultant force using vector addition, and the magnitude of the resultant can be found using the diagonal of the parallelogram.

Step 2: Apply the Law of Cosines

Use the law of cosines to calculate the magnitude of the resultant force \( R \). The formula for the magnitude of the resultant is:\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(\theta)} \]where \( F_1 = 12 \) pounds, \( F_2 = 18 \) pounds, and \( \theta = 75^{\circ} \).

Step 3: Calculate \( \cos(75^{\circ}) \)

Find the value of \( \cos(75^{\circ}) \) using a calculator or trigonometric tables: \[ \cos(75^{\circ}) \approx 0.2588 \].

Step 4: Substitute the Values into the Formula

Substitute the known values into the law of cosines formula:\[ R = \sqrt{12^2 + 18^2 + 2 \times 12 \times 18 \times 0.2588} \].

Step 5: Perform the Calculations

Calculate each term inside the square root:- \( 12^2 = 144 \)- \( 18^2 = 324 \)- \( 2 \times 12 \times 18 \times 0.2588 \approx 111.6192 \)Combine these values:\[ R = \sqrt{144 + 324 + 111.6192} = \sqrt{579.6192} \].

Step 6: Find the Resultant Force

Calculate the square root to find the magnitude of the resultant force:\[ R \approx 24.07 \].

Step 7: Round the Result

The magnitude of the resultant force, rounded to the nearest hundredth pound, is approximately 24.07 pounds.

Key Concepts

Vector Addition

Vector addition is a fundamental concept when dealing with forces. Imagine two horses pulling a cart in different directions. The effect they have on the cart depends on both their directions and strengths. When forces are vectors, their magnitudes and directions determine their sum. This process is referred to as "vector addition."

• Each vector represents a force with a specific direction and magnitude.
• Using vector addition, we can combine multiple forces into a single, equivalent force.

When the forces do not act along the same line, we use geometric methods or vector mathematics to find the resultant vector. This is particularly useful for calculating the overall effect of forces acting in various directions, such as in our exercise.

Law of Cosines

The Law of Cosines is a crucial tool in solving problems involving vectors, especially when dealing with non-right angles. It is an extension of the Pythagorean theorem, which works for any triangle, not just those with a right angle.
For two sides of a triangle and the included angle, the Law of Cosines states:

• \[ c^2 = a^2 + b^2 - 2ab \, \cos \theta \]

In our force problem, the sides are given by the magnitudes of the vectors, and the included angle is the angle between them. By substituting known values into the equation, we simplify to find the length of the resultant force vector (the diagonal of the parallelogram in the exercise). This useful law allows us to solve for unknowns even when working with oblique angles.

Trigonometry

Trigonometry provides valuable tools in understanding relations in triangles and vectors. In this case, we specifically use cosine to find relationships between the forces.

• Cosine helps determine how much forces influence each other.
• Useful trigonometric values like \( \cos(75^{\circ}) \) are found using calculators or tables.

By finding \( \cos(75^{\circ}) \approx 0.2588 \), we can calculate how much the forces contribute to the resultant force in a specified direction. Trigonometric identities and functions allow us to solve complex real-world problems like the one in the exercise efficiently.

Parallelogram Rule

The Parallelogram Rule is a geometric method for visualizing and calculating the resultant force when two vectors act simultaneously. When two vectors are placed tail-to-tail, they form adjacent sides of a parallelogram.

• The diagonal of the parallelogram represents the resultant vector.
• It explains how combined effects arise from individual components.

Applying this rule in vector addition helps transform abstract computations into a more visual understanding. For the problem at hand, this rule shows why solving for the parallelogram's diagonal provides the resultant force's magnitude. Understanding this rule helps us intuitively grasp how different forces combine to produce a single effect.