Problem 22 - Fundamentals of Physics

Question

(a) What is the sum of the following four vectors in unitvector notation? For that sum, what are (b) the magnitude, (c) the angle in degrees, and (d) the angle in radians? $$\begin{array}{ll}\vec{E}: 6.00 \mathrm{~m} \mathrm{at}+0.900 \mathrm{rad} & \vec{F}: 5.00 \mathrm{~m} \text { at }-75.0^{\circ} \\\\\vec{G}: 4.00 \mathrm{~m} \mathrm{at}+1.20 \mathrm{rad} & \vec{H}: 6.00 \mathrm{~m} \text { at }-210^{\circ}\end{array}$$

Step-by-Step Solution

Answer

The sum is \(1.31 \hat{i} + 6.67 \hat{j}\) m with magnitude \(6.80\) m, angle \(78.87^\circ\) or \(1.38\) rad.

Step 1: Convert degrees to radians for Vector F and Vector H

Vector F is at \(-75.0^\circ\). Convert this to radians using the formula \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\) which gives \(-1.309\) rad. Vector H is at \(-210^\circ\), so in radians this is \(-3.665\) rad.

Step 2: Determine components of vectors

For each vector, determine the x-component using \(x = r \cos(\theta)\) and the y-component using \(y = r \sin(\theta)\). For vector \(\vec{E}\), x-component is \(6.00 \cos(0.900) = 3.77\) m and y-component is \(6.00 \sin(0.900) = 4.77\) m. For vector \(\vec{F}\), x-component is \(5.00 \cos(-1.309) = 1.29\) m and y-component is \(5.00 \sin(-1.309) = -4.83\) m. For vector \(\vec{G}\), x-component is \(4.00 \cos(1.20) = 1.45\) m and y-component is \(4.00 \sin(1.20) = 3.73\) m. For vector \(\vec{H}\), x-component is \(6.00 \cos(-3.665) = -5.20\) m and y-component is \(6.00 \sin(-3.665) = 3.00\) m.

Step 3: Sum components to find resultant vector in unit-vector notation

Add all the x-components and y-components of vectors E, F, G, and H: - Summing x-components: \(3.77 + 1.29 + 1.45 - 5.20 = 1.31\) m - Summing y-components: \(4.77 - 4.83 + 3.73 + 3.00 = 6.67\) m The resultant vector \(\vec{R} = 1.31 \hat{i} + 6.67 \hat{j}\) m.

Step 4: Calculate the magnitude of the resultant vector

The magnitude of vector \(\vec{R}\) is calculated using Pythagorean theorem: \(\|\vec{R}\| = \sqrt{(1.31)^2 + (6.67)^2}\)This results in \(\|\vec{R}\| \approx 6.80\) meters.

Step 5: Determine the angle in degrees and radians

Calculate the angle \(\theta\) using the arctangent function: \(\theta = \tan^{-1}\left(\frac{6.67}{1.31}\right)\)This results in \(\theta \approx 78.87^\circ\). Convert this angle to radians using the conversion \(\theta \times \frac{\pi}{180} \approx 1.38\) radians.

Key Concepts

Unit Vector Notation

When dealing with vector addition, we often work with unit vector notation as it's a precise way to represent vectors. In unit vector notation, every vector is expressed as a combination of its components along the x-axis and y-axis, typically using \(\hat{i}\) for the x-direction and \(\hat{j}\) for the y-direction. For example, a vector \((a\hat{i} + b\hat{j})\) clearly tells us that 'a' is the magnitude along the x-axis and 'b' is the magnitude along the y-axis.
This notation becomes particularly handy when adding vectors, as you can simply sum the respective components to find the resultant vector. For instance, if adding multiple vectors, you add up all the x-components to get the resultant x-component, and similarly sum all the y-components for the resultant y-component.

• Express vectors in the form of \(x\hat{i} + y\hat{j}\)
• Add corresponding components to find the resultant vector

In the given problem, after computing the components of each vector, we found the sum: \(\vec{R} = 1.31\hat{i} + 6.67\hat{j}\) m.

Magnitude of a Vector

The magnitude of a vector provides insight into how 'long' or 'strong' that vector is, essentially, it is the vector’s length irrespective of direction. To calculate the magnitude, you use the Pythagorean theorem, which involves the vector’s components.
The formula is straightforward: if a vector \(\vec{v}\) is expressed as \(x\hat{i} + y\hat{j}\), its magnitude \(\|\vec{v}\|\) is found using:\[\|\vec{v}\| = \sqrt{x^2 + y^2}\]In the problem, for our resultant vector \(\vec{R} = 1.31\hat{i} + 6.67\hat{j}\), the magnitude becomes \(\|\vec{R}\| = \sqrt{(1.31)^2 + (6.67)^2}\).
Calculating this, we find \(\|\vec{R}\| \approx 6.80\) meters.

• Magnitude tells us the size/strength of the vector.
• Use the Pythagorean theorem for calculation.

Understanding magnitude is crucial as it has practical applications in determining things like speed in physics, where the vector represents velocity.

Angle Conversion

Converting between degrees and radians is an essential skill in physics and mathematics because different contexts may require different units. Degrees are more intuitive for daily use, while radians are the standard unit of angular measure in mathematical calculations because of their direct relationship to the arc length.

To convert degrees to radians, multiply the degree measurement by \(\frac{\pi}{180}\). This stems from the fact that \(360\degree\) equals \(2\pi\) radians. Hence the formula: \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]In the exercise, converting \(-75.0^\circ\) for Vector \(\vec{F}\) gives approximately \(-1.309\) radians, and \(-210^\circ\) for Vector \(\vec{H}\) converts to \(-3.665\) radians.

• Radians are more natural for calculations involving pi and arc length.
• Always remember that \(180^\circ = \pi\) radians.

Understanding angle conversions is crucial when working with trigonometric functions in physics and engineering.