91Ó°ÊÓ

Force vectors are a fundamental concept in physics that describe the direction and magnitude of forces acting on an object. In this exercise, we have two force vectors, \( \overrightarrow{\mathbf{F}}_1 \) and \( \overrightarrow{\mathbf{F}}_2 \), which have the same magnitude but different directions.

A vector can be visualized as an arrow, where the length represents the force's strength, and the direction of the arrow shows where the force is headed. When dealing with vectors, it's crucial to consider both magnitude and direction.

• Magnitude tells us how strong the force is.
• Direction informs us about the path along which the force acts.
  

For a sailboat subjected to these forces, the force vector \( \overrightarrow{\mathbf{F}}_1 \) acts at an angle west of due north. This means it's not pointing straight along the boat's path but instead a bit off to the side. The force vector \( \overrightarrow{\mathbf{F}}_2 \), however, points directly due north, perfectly aligned with the direction of travel, making it straightforward to analyze when calculating work.

Calculating the angle between the force direction and the direction of movement is important when you want to understand how effective a force is in doing work. In this problem, the angle \( \theta \) is the one between \( \overrightarrow{\mathbf{F}}_1 \) and due north, the path along which the boat is moving.

This angle affects how much of the force actually contributes to doing work on the object. Only the component of the force that is aligned with the movement direction does productive work.

• To find this angle, we use the formula for work: \( W = F \cdot d \cdot \cos(\theta) \), where \( \theta \) is the angle between the force and the movement.
• The problem states that the work done by both forces over their respective distances is equal, allowing us to set up a comparison equation.
• Solving this, we found that \( \cos(\theta) = \frac{47}{52} \). This ratio shows the extent to which \( \overrightarrow{\mathbf{F}}_1 \) is aligned with the direction of motion.
  

The cosine function is a vital mathematical tool in physics, especially when dealing with vector analysis and work. It helps in determining how much of a force contributes to motion in a particular direction by relating the angle to the force's effectiveness.

Using \( \cos(\theta) \), we measure the alignment of a force with the direction of motion. A \( \cos(\theta) \) of 1 means perfect alignment, while a value of 0 indicates a force perpendicular to the direction of movement, contributing nothing to work.

• In this problem, we see that \( \cos(\theta) \) is approximately \( \frac{47}{52} \), which is nearly 0.9. This indicates a force component mostly aligned along the path.
• By understanding \( \cos(\theta) \), we get crucial information about how the angle affects work. In this case, even with a 25.23 degree offset, much of \( \overrightarrow{\mathbf{F}}_1 \) helps move the sailboat northward.
• The angle is computed by taking the inverse cosine, \( \theta = \cos^{-1}\left(\frac{47}{52}\right) \), providing the precise degree to which the force diverges from the direct path.
  

This mathematical operation helps us solve real-world problems where force directions aren't perfectly aligned with movement, illustrating the power and practicality of trigonometry in physics.