A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.


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Verified step by step guidance

1. Convert the angles from degrees and minutes to decimal degrees for easier calculations. For example, 42° 10′ becomes 42 + 10/60 degrees.
2. Use the Law of Cosines to find the magnitude of the resultant force. The formula is: \( R^2 = F_1^2 + F_2^2 - 2F_1F_2 \cos(\theta) \), where \( R \) is the resultant force, \( F_1 \) and \( F_2 \) are the magnitudes of the two forces, and \( \theta \) is the angle between them.
3. Apply the Law of Sines to find the magnitude of the second force. The formula is: \( \frac{F_2}{\sin(\alpha)} = \frac{R}{\sin(\beta)} \), where \( \alpha \) is the angle opposite \( F_2 \) and \( \beta \) is the angle opposite the resultant force.
4. Calculate the angle \( \alpha \) using the given angles and the fact that the sum of angles in a triangle is 180 degrees. \( \alpha = 180° - (\text{angle between the forces} + \text{angle of the resultant with the first force}) \).
5. Substitute the known values into the Law of Sines equation to solve for the magnitude of the second force.

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this context, forces are represented as vectors, and their magnitudes and directions must be considered. The resultant vector is found using methods such as the parallelogram law or trigonometric functions, which help in calculating the magnitude and direction of the combined forces.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. These functions are essential for resolving vectors into their components and for calculating angles and magnitudes in problems involving forces. In this question, they will be used to find the components of the second force and the resultant force based on the given angles.

Law of Sines and Cosines

The Law of Sines and the Law of Cosines are fundamental principles used to solve triangles when dealing with non-right triangles. The Law of Sines relates the ratios of the lengths of sides to the sines of their opposite angles, while the Law of Cosines provides a way to calculate a side or angle when two sides and the included angle are known. These laws will be crucial in determining the magnitudes of the second force and the resultant in this problem.