What are the x- and y-components of the velocity vector shown in FIGURE EX3.20?

검증된 단계별 안내

Step 1: Identify the given information from the diagram. The force vector has a magnitude of 25 N and is directed at an angle of 20° from the positive x-axis. The goal is to find the x- and y-components of this vector.

Step 2: Recall the formulas for resolving a vector into its components. The x-component of the vector is given by \( F_x = F \cdot \cos(\alpha) \), and the y-component is given by \( F_y = F \cdot \sin(\alpha) \), where \( F \) is the magnitude of the vector and \( \alpha \) is the angle it makes with the positive x-axis.

Step 3: Substitute the given values into the formulas. For the x-component: \( F_x = 25 \cdot \cos(20°) \). For the y-component: \( F_y = 25 \cdot \sin(20°) \).

Step 4: Use a calculator or trigonometric tables to find the values of \( \cos(20°) \) and \( \sin(20°) \). These values will be multiplied by the magnitude of the force (25 N) to determine the components.

Step 5: Interpret the results. The x-component represents the horizontal part of the vector along the x-axis, and the y-component represents the vertical part of the vector along the y-axis. Ensure the signs of the components are consistent with the direction of the vector in the diagram.

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주요 개념

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Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. In two dimensions, any vector can be broken down into its x (horizontal) and y (vertical) components using trigonometric functions. This is essential for analyzing forces, as it allows us to understand how much of the force acts in each direction.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. For a vector at an angle, the x-component can be found using the cosine of the angle, while the y-component is found using the sine. In this case, with a 20-degree angle, these functions will help determine the respective components of the 25 N force.

Force and Direction

Force is a vector quantity that has both magnitude and direction. In this problem, the force of 25 N is directed at an angle of 20 degrees from the x-axis, which indicates that it has both an upward (y) and a horizontal (x) component. Understanding the direction of the force is crucial for accurately calculating its components and analyzing the overall effect of the force in a given scenario.