Problem 40 - College Physics

Question

\(\cdot\) In each of the cases that follow, the magnitude of a vector is given along with the counterclockwise angle it makes with the \(+x\) axis. Use trigonometry to find the \(x\) and \(y\) components of the vector. Also, sketch each vector approximately to scale to see if your calculated answers seem reasonable. (a) 50.0 \(\mathrm{N}\) at \(60.0^{\circ},(\mathrm{b}) 75 \mathrm{m} / \mathrm{s}\) at \(5 \pi / 6 \mathrm{rad},(\mathrm{c}) 254 \mathrm{lb}\) at \(325^{\circ},\) (d) 69 \(\mathrm{km}\) at 1.1\(\pi \mathrm{rad} .\)

Step-by-Step Solution

Answer

(a) (25.0 N, 43.3 N), (b) (-64.95 m/s, 37.5 m/s), (c) (208.29 lb, -147.06 lb), (d) (-27.68 km, -63.91 km)

Step 1: Understanding Vector Components

Vectors can be broken down into their horizontal (x-axis) and vertical (y-axis) components using trigonometry. Given a vector's magnitude and angle, we use the sine and cosine functions to determine these components.

Step 2: Converting Angles to Radians If Needed

If angles are given in degrees, convert them to radians for consistency, especially since some are already provided in radians. Use the formula for conversion: \[\theta\text{ (rad)} = \theta\text{ (degrees)} \times \frac{\pi}{180}\]

Step 3: Calculating Components for Vector (a)

For vector (a): magnitude \( = 50.0 \mathrm{N}\), angle \( = 60.0^{\circ}\).Convert \(60.0^{\circ}\) to radians: \[60.0^{\circ} \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}\]Calculate \(x\) and \(y\) components:\[x = 50.0 \cdot \cos\left(\frac{\pi}{3}\right) = 50.0 \cdot 0.5 = 25.0 \mathrm{N}\]\[y = 50.0 \cdot \sin\left(\frac{\pi}{3}\right) = 50.0 \cdot \frac{\sqrt{3}}{2} \approx 43.3 \mathrm{N}\]

Step 4: Calculating Components for Vector (b)

For vector (b): magnitude \( = 75 \mathrm{m/s}\), angle \( = \frac{5\pi}{6} \text{ rad}\).Calculate \(x\) and \(y\) components:\[x = 75 \cdot \cos\left(\frac{5\pi}{6}\right) = 75 \cdot \left(-\frac{\sqrt{3}}{2}\right) \approx -64.95 \mathrm{m/s}\]\[y = 75 \cdot \sin\left(\frac{5\pi}{6}\right) = 75 \cdot \frac{1}{2} = 37.5 \mathrm{m/s}\]

Step 5: Calculating Components for Vector (c)

For vector (c): magnitude \( = 254 \mathrm{lb}\), angle \( = 325^{\circ}\).Convert \(325^{\circ}\) to radians:\[325^{\circ} \times \frac{\pi}{180} \approx 5.672\text{ rad}\]Calculate \(x\) and \(y\) components:\[x = 254 \cdot \cos(5.672) \approx 208.29 \mathrm{lb}\]\[y = 254 \cdot \sin(5.672) \approx -147.06 \mathrm{lb}\]

Step 6: Calculating Components for Vector (d)

For vector (d): magnitude \( = 69 \mathrm{km}\), angle \( = 1.1\pi \text{ rad}\).Calculate \(x\) and \(y\) components:\[x = 69 \cdot \cos(1.1\pi) \approx -27.68 \mathrm{km}\]\[y = 69 \cdot \sin(1.1\pi) \approx -63.91 \mathrm{km}\]

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In the context of vector components, trigonometry helps us break down vectors into their basic parts: the horizontal and vertical components. Given a vector's magnitude and the angle it forms with the positive x-axis, we can find its components using the cosine and sine functions, respectively.
For any vector that makes an angle \( \theta \) with the x-axis:

• The x-component (horizontal) is given by \( x = \text{magnitude} \times \cos(\theta) \).
• The y-component (vertical) is given by \( y = \text{magnitude} \times \sin(\theta) \).

This method allows us to translate complex diagonal forces or movements into simpler, more manageable forms.

Magnitude and Direction

Understanding the magnitude and direction of a vector is crucial to finding its components. Magnitude refers to the size or length of the vector, often representing distance or strength. Meanwhile, direction is the angle that the vector makes with a reference axis, typically the positive x-axis.

• Magnitude is simply the scalar quantity and is always non-negative.
• Direction is typically measured in degrees or radians, indicating which way the vector points.

Visualizing a vector's direction involves drawing the vector as an arrow from the origin. The angle can be measured counterclockwise from the positive x-axis, which helps to standardize our calculations.

Unit Conversion

Unit conversion is an essential skill when dealing with vectors, especially when angles are provided in different units, such as degrees and radians. In vector calculations, consistency in units allows for more accurate results.
When angles need conversion, use the following formula:

• Convert degrees to radians: \[ \theta\text{ (rad)} = \theta\text{ (degrees)} \times \frac{\pi}{180} \]

This formula helps maintain consistency, particularly when vectors are compared or combined, ensuring calculations remain correct.

Vector Sketching

Vector sketching is a valuable technique that provides a visual representation of vectors and their components. Sketching a vector helps verify the logical placement of its x and y components.
To sketch a vector approximately:

• Begin at the origin point (where x and y axes cross).
• Draw an arrow in the direction specified by the angle.
• Use the magnitude to set the length of the arrow.
• Break the vector into two arrows, parallel to the x-axis and y-axis, representing the vector's components.

_Approximately sketching_ vectors helps cross-check calculations, providing a reference to ensure the numeric answers align with the visual direction and scale.