Problem 2 - Fundamentals of Physics

Question

A displacement vector \(\vec{r}\) in the \(x y\) plane is \(15 \mathrm{~m}\) long and directed at angle \(\theta=30^{\circ}\) in Fig. 3 -26. Determine (a) the \(x\) component and (b) the \(y\) component of the vector.

Step-by-Step Solution

Answer

The x-component is approximately 12.99 m, and the y-component is 7.5 m.

Step 1: Understanding the Problem

We have a vector \(\vec{r}\) that represents displacement in the \(xy\) plane, which has a magnitude of 15 m and is directed at an angle of 30 degrees from the positive x-axis. We need to find the components of this vector along the x-axis and y-axis.

Step 2: Using the Cosine Function for x-component

We know that the x-component \(r_x\) of a vector can be found using the cosine of the angle: \(r_x = r \cdot \cos(\theta)\). Here, \(r = 15 \text{ m}\) and \(\theta = 30^\circ\). So, substitute the known values into the formula: \[r_x = 15 \cdot \cos(30^\circ)\].

Step 3: Calculating the x-component

Calculate \(\cos(30^\circ)\). From trigonometry, we know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\). Therefore: \[r_x = 15 \cdot \frac{\sqrt{3}}{2} = \frac{15\sqrt{3}}{2} \approx 12.99 \text{ m}\].

Step 4: Using the Sine Function for y-component

The y-component \(r_y\) of a vector can be found using the sine of the angle: \(r_y = r \cdot \sin(\theta)\). We substitute the values \(r = 15 \text{ m}\) and \(\theta = 30^\circ\) into the formula: \[r_y = 15 \cdot \sin(30^\circ)\].

Step 5: Calculating the y-component

Calculate \(\sin(30^\circ)\). We know from trigonometry that \(\sin(30^\circ) = \frac{1}{2}\). Therefore: \[r_y = 15 \cdot \frac{1}{2} = 7.5 \text{ m}\].

Key Concepts

Vector Components

Understanding vector components is crucial for solving problems involving vectors, such as displacement vectors in the xy plane. When we talk about vector components, we break down any vector into its parts along the coordinate axes. In this case, we have an x-axis component and a y-axis component of the displacement vector.

• The x-component: This tells us how much of the vector's magnitude points along the x-axis.
• The y-component: This indicates how much of the vector lies along the y-axis.

The beauty of using vector components is they allow us to work with smaller, simpler pieces of a vector. This decomposition transforms solving vector problems into straightforward arithmetic using basic trigonometric functions. It's essential for physics and engineering, helping us to analyze movements and forces more effectively.

Trigonometric Functions

To find the x and y components of a vector, trigonometric functions come to the rescue. These functions relate the angles and sides of a triangle, which in the case of vectors, helps us determine the components using the vector's angle relative to the coordinate axes.

• Cosine (\(\cos\)) Function: Used to calculate the x-component. It helps us find how much of the vector's magnitude aligns horizontally with the x-axis. It's derived from adjacent side over hypotenuse in the context of right triangles.
• Sine (\(\sin\)) Function: Needed for the y-component. It calculates the vertical part of the vector's magnitude. In triangles, it's the ratio of the opposite side to the hypotenuse.

When applying these functions to our vector problem, we use a known angle and the vector's magnitude to solve for each component separately. This mathematical process is fundamental in vector decomposition.

Coordinate Systems

Coordinate systems are the frameworks that help us define the position of points or vectors in space. In problems involving vectors, such as the displacement vector in the exercise, using a coordinate system allows for a clear representation of the vector's direction and magnitude in a two-dimensional plane.

• 2D Cartesian Coordinate System: It's composed of two perpendicular axes - the x-axis (horizontal) and the y-axis (vertical). The intersection of these axes forms the origin, the starting point for positioning objects in this plane.
• Direction and Angle: In our problem, the vector is described using a directional angle (30 degrees from the positive x-axis), which sets the stage for applying trigonometric functions to find components.

By understanding the coordinate system, we can use angles and vector magnitudes effectively to analyze direction and magnitude. This makes breaking down vectors and solving their components intuitive and practical.