Let B→\overrightarrow{B} = (5.0 m, 30 degrees counterclockwise from vertically up). Find the x- and y-components of B→\overrightarrow{B} in each of the two coordinate systems shown in FIGURE EX3.21.

Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

문제 21a

3장, 문제 21a

Let $\vec{B}$ = (5.0 m, 30 degrees counterclockwise from vertically up). Find the x- and y-components of $\vec{B}$ in each of the two coordinate systems shown in FIGURE EX3.21.

검증된 단계별 안내

Step 1: Identify the vector B = (5.0 m, 30 degrees counterclockwise from vertically up). In the normal coordinate system, the vertical axis is the y-axis, and the horizontal axis is the x-axis. In the tilted coordinate system, the axes are rotated by 15 degrees counterclockwise.

Step 2: For the normal coordinate system, resolve the vector B into its x- and y-components using trigonometric functions. The x-component is given by B_x = B * sin(θ), and the y-component is given by B_y = B * cos(θ), where θ = 30 degrees.

Step 3: For the tilted coordinate system, adjust the angle of the vector relative to the tilted axes. The angle of B relative to the tilted y-axis is θ_tilted = 30 degrees - 15 degrees = 15 degrees. Use the same trigonometric functions to resolve the vector: B_x_tilted = B * sin(θ_tilted) and B_y_tilted = B * cos(θ_tilted).

Step 4: Substitute the magnitude of B (5.0 m) and the angles (30 degrees for normal, 15 degrees for tilted) into the trigonometric equations to calculate the components. Ensure you use the correct trigonometric values for sin and cos of the angles.

Step 5: Interpret the results. The components in the normal coordinate system will align with the standard x- and y-axes, while the components in the tilted coordinate system will align with the rotated axes. This demonstrates how vector components depend on the orientation of the coordinate system.

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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. For a vector B, its components can be calculated using trigonometric functions: the x-component is found using Bx = B * cos(θ) and the y-component using By = B * sin(θ), where θ is the angle the vector makes with the reference axis.

Coordinate Systems

Coordinate systems provide a framework for describing the position and direction of vectors. In this problem, two systems are used: a normal Cartesian system and a tilted system, which is rotated by 15 degrees. The choice of coordinate system affects how the components of the vector are calculated, as the angle θ will differ based on the orientation of the axes.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. They are essential for resolving vectors into components. In this context, they help determine how much of the vector B lies in the x and y directions, depending on the angle of the vector relative to the coordinate axes.