Force Vectors and Coplanar Force Systems

Flashcards covering vector operations, scalar vs. vector properties, Cartesian vector notation, coplanar force addition, and resolution of vectors based on Statics by R.C. Hibbeler.

What distinguishes a scalar quantity from a vector quantity?

A scalar quantity has only magnitude (positive or negative) and adds using simple arithmetic, whereas a vector quantity has both magnitude and direction and adds using the parallelogram law.

According to the Reading Quiz, which law must be used for vector addition?

The parallelogram law.

What does the resolution of a vector mean in statics?

Breaking up a vector into its individual components, which is effectively using the parallelogram law in reverse.

What unit vectors are used to designate the $x$ and $y$ axes in Cartesian vector notation?

The unit vectors $\mathbf{i}$ for the $x$-axis and $\mathbf{j}$ for the $y$-axis.

What are the three steps required to add a system of coplanar forces using Cartesian vector notation?

Step 1: Resolve each force into its $x$ and $y$ components. Step 2: Add all $x$-components together and all $y$-components together to find the resultant components $F_{Rx}$ and $F_{Ry}$. Step 3: Calculate the magnitude and angle of the resultant vector.

What formula is used to calculate the magnitude of a 2-D resultant vector $F_R$ from its rectangular components $F_{Rx}$ and $F_{Ry}$?

$$F_R = \sqrt{F_{Rx}^2 + F_{Ry}^2}$$

What formula is used to determine the direction angle \theta of a resultant force vector from its rectangular components $F_{Rx}$ and $F_{Ry}$?

$$\theta = \tan^{-1}\left|\frac{F_{Ry}}{F_{Rx}}\right|$$

For the three concurrent forces acting on the tent post shown, what are the Cartesian vector components of $F_1$, $F_2$, and $F_3$?

$F_1 = \{0\mathbf{i} + 300\mathbf{j}\}\,\text{N}$, $F_2 = \{-450 \cos(45^\circ)\mathbf{i} + 450 \sin(45^\circ)\mathbf{j}\}\,\text{N} = \{-318.2\mathbf{i} + 318.2\mathbf{j}\}\,\text{N}$, and $F_3 = \left\{\left(\frac{3}{5}\right)600\mathbf{i} + \left(\frac{4}{5}\right)600\mathbf{j}\right\}\,\text{N} = \{360\mathbf{i} + 480\mathbf{j}\}\,\text{N}$.

What is the magnitude and directional angle \phi of the resultant force for the tent post problem with components $\mathbf{F}_R = \{41.80\mathbf{i} + 1098\mathbf{j}\}\,\text{N}$?

Magnitude $F_R = \sqrt{(41.80)^2 + (1098)^2} = 1099\,\text{N}$ and direction angle $\phi = \tan^{-1}\left(\frac{1098}{41.80}\right) = 87.8^\circ$.

What is the Law of Sines formula used for determining the magnitudes of vector components in a triangle?

$$\frac{A}{\sin(a)} = \frac{B}{\sin(b)} = \frac{C}{\sin(c)}$$

For the $30\,\text{lb}$ force acting on the pipe, what are the resolved component magnitudes $F_u$ and $F_v$ along the non-perpendicular $u$ and $v$ axes?

$F_u = 22.0\,\text{lb}$ and $F_v = 15.5\,\text{lb}$, determined using the law of sines: $\frac{30}{\sin(105^\circ)} = \frac{F_u}{\sin(45^\circ)} = \frac{F_v}{\sin(30^\circ)}$.

For the three concurrent forces acting on the bracket, what are the Cartesian vector forms of $F_1$, $F_2$, and $F_3$?

$F_1 = \{680\mathbf{i} - 510\mathbf{j}\}\,\text{N}$, $F_2 = \{-312.5\mathbf{i} - 541.3\mathbf{j}\}\,\text{N}$, and $F_3 = \{-530.3\mathbf{i} + 530.3\mathbf{j}\}\,\text{N}$.

What is the resultant vector $\mathbf{F}_R$, its magnitude, and its angle \theta measured counterclockwise from the positive $x$ axis for the bracket group problem?

Resultant vector $\mathbf{F}_R = \{-162.8\mathbf{i} - 520.9\mathbf{j}\}\,\text{N}$, magnitude $F_R = 546\,\text{N}$, directional angle in 3rd quadrant $\phi = 72.6^\circ$, and angle from the positive $x$-axis $\theta = 253^\circ$.

How is a force vector $F = 80\,\text{N}$ directed at an angle of $30^\circ$ below the positive $x$-axis expressed in Cartesian vector form?

$$F = \{80 \cos(30^\circ)\mathbf{i} - 80 \sin(30^\circ)\mathbf{j}\}\,\text{N}$$

What is the magnitude of the resultant force $\mathbf{F}_1 + \mathbf{F}_2$ when $\mathbf{F}_1 = \{10\mathbf{i} + 20\mathbf{j}\}\,\text{N}$ and $\mathbf{F}_2 = \{20\mathbf{i} + 20\mathbf{j}\}\,\text{N}$?

$50\,\text{N}$, calculated from $\mathbf{F}_R = \{30\mathbf{i} + 40\mathbf{j}\}\,\text{N}$ as $\sqrt{30^2 + 40^2} = 50\,\text{N}$.