Problem 47 - Fundamentals of Physics

Question

Vector \(\vec{a}\) has a magnitude of \(5.0 \mathrm{~m}\) and is directed east. Vector \(\vec{b}\) has a magnitude of \(4.0 \mathrm{~m}\) and is directed \(35^{\circ}\) west of due north. What are (a) the magnitude and (b) the direction of \(\vec{a}+\vec{b}\) ? What are (c) the magnitude and (d) the direction of \(\vec{b}-\vec{a} ?\) (e) Draw a vector diagram for each combination.

Step-by-Step Solution

Answer

Magnitude of \( \vec{a} + \vec{b} \) is 4.23 m at 50.4° north of east. Magnitude of \( \vec{b} - \vec{a} \) is 7.96 m at 24.1° north of west.

Step 1: Understanding vector components

Vector \( \vec{a} \) is purely in the east direction, so its components are \( a_x = 5.0 \text{ m} \) and \( a_y = 0 \text{ m} \). For vector \( \vec{b} \), having a direction of 35° west of north means \( \theta = 90° + 35° = 125° \) relative to the positive x-axis (east direction). Use \( b_x = 4.0 \cos(125°) \) and \( b_y = 4.0 \sin(125°) \) to find components of \( \vec{b} \).

Step 2: Calculate vector components

Using trigonometry and the given angles, calculate: \( b_x = 4.0 \cos(125°) = -2.30 \text{ m} \), \( b_y = 4.0 \sin(125°) = 3.28 \text{ m} \). Vector \( \vec{a} = (5.0, 0) \text{ m} \) and vector \( \vec{b} = (-2.30, 3.28) \text{ m} \).

Step 3: Calculate \( \vec{a} + \vec{b} \) components

Add the components: \((\vec{a} + \vec{b})_x = a_x + b_x = 5.0 - 2.30 = 2.70 \text{ m} \) and \((\vec{a} + \vec{b})_y = a_y + b_y = 0 + 3.28 = 3.28 \text{ m} \).

Step 4: Magnitude of \( \vec{a} + \vec{b} \)

Find the magnitude using the Pythagorean theorem: \( |\vec{a} + \vec{b}| = \sqrt{(2.70)^2 + (3.28)^2} = \sqrt{7.29 + 10.76} \approx 4.23 \text{ m} \).

Step 5: Direction of \( \vec{a} + \vec{b} \)

Calculate the direction angle \( \phi \) from the x-axis using \( \tan \phi = \frac{3.28}{2.70} \): \( \phi = \tan^{-1}(1.215) \approx 50.4° \) north of east.

Step 6: Calculate \( \vec{b} - \vec{a} \) components

Subtract the components: \((\vec{b} - \vec{a})_x = b_x - a_x = -2.30 - 5.0 = -7.30 \text{ m} \) and \((\vec{b} - \vec{a})_y = b_y - a_y = 3.28 - 0 = 3.28 \text{ m} \).

Step 7: Magnitude of \( \vec{b} - \vec{a} \)

Find the magnitude using the Pythagorean theorem: \( |\vec{b} - \vec{a}| = \sqrt{(-7.30)^2 + (3.28)^2} = \sqrt{53.29 + 10.76} \approx 7.96 \text{ m} \).

Step 8: Direction of \( \vec{b} - \vec{a} \)

Calculate the direction angle \( \theta \) from the x-axis using \( \tan \theta = \frac{3.28}{-7.30} \): \( \theta = \tan^{-1}(-0.449) \approx -24.1° \) or \( 24.1° \) north of west.

Step 9: Draw vector diagrams

Sketch two diagrams: one showing vectors \( \vec{a} \) and \( \vec{b} \) and their resultant \( \vec{a} + \vec{b} \); the second showing \( \vec{a} \) reflected and subtracted as \( \vec{b} - \vec{a} \). Both diagrams use arrows and angle markers to show direction and magnitude.

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to determine a resultant vector. To find the resultant vector, you add the corresponding components of each vector.
For example, consider vectors \( \vec{a} \) and \( \vec{b} \). If vector \( \vec{a} \) is directed east with components \( (a_x, a_y) \) and vector \( \vec{b} \) is inclined at an angle, transforming it into components \( (b_x, b_y) \), their sum is given by:

• \((\vec{a} + \vec{b})_x = a_x + b_x\)
• \((\vec{a} + \vec{b})_y = a_y + b_y\)

The resultant vector \( \vec{a} + \vec{b} \) would then be expressed as the sum of these new components.
This creates a new vector whose direction and magnitude we calculate using trigonometry. The graphical method often visualizes this as 'tip-to-tail' where the tail of one vector connects to the tip of another.

Vector Subtraction

Vector subtraction is remarkably similar to vector addition, but it involves reversing the direction of the vector being subtracted. This is useful in determining the resultant of two forces acting in opposite directions.
Take vectors \( \vec{b} \) and \( \vec{a} \), the subtraction \( \vec{b} - \vec{a} \) first requires you to reverse \( \vec{a} \) (changing all its components signs), and then add:

• \((\vec{b} - \vec{a})_x = b_x - a_x\)
• \((\vec{b} - \vec{a})_y = b_y - a_y\)

The subtraction technique involves being clear about the direction as vectors point in specific orientations in space. In physics problems, this is absolutely key in correctly identifying the direction of forces or displacements.

Magnitude and Direction

The magnitude of a vector represents its length, which is a scalar quantity. To determine it, typically the Pythagorean theorem is employed. For a vector with components \( (x, y) \), the magnitude \( |\vec{v}| \) is:
\[ |\vec{v}| = \sqrt{x^2 + y^2} \]
The direction is an angle that indicates where the vector points. Calculated as the angle \( \theta \) from a reference direction, often the positive x-axis, the expression is:

• \( \theta = \tan^{-1}\left(\frac{y}{x}\right)\)

These calculations provide a complete description of the vector, necessary for understanding how it fits into a physical scenario.

Trigonometry in Physics

Trigonometry is critical in physics, especially when dealing with vectors. With vectors not always aligned with the axes, breaking them into components using sine and cosine becomes essential.

When a vector \( \vec{b} \) has a magnitude and an angle \( \theta \) to a reference axis, its components are:

• \( b_x = b \cos(\theta) \)
• \( b_y = b \sin(\theta) \)

These formulas allow conversion between polar and rectangular (Cartesian) forms of vectors, facilitating addition, subtraction, and analysis of vectors.
Such trigonometric practice enables concise solutions in physics problems, including calculating forces, displacements, and various physical phenomena.