Statics/Forces As Vectors

Reading(s):

Vectors, Chapter 1.1 - 1.7

Scalar
A scalar is a quantity possessing only a magnitude. Examples include mass, volume, and length. In this book, scalars are represented by letters in italic type: $$A$$. Scalar quantities may be manipulated following the rules of simple algebra.

Vector
A vector is a quantity that has both a magnitude and a direction. Examples include velocity, position, and force. In this book, vectors are represented by letters with arrows over them: $$\vec A$$. Vector quantities are manipulated using vector mathematics, which is described in some detail in the following section.

Velocity vs. Speed
Consider a car traveling South at a speed of 110 km per hour.

We can describe the motion of the car as a velocity with a magnitude of 110 km per hour and a direction of south. The velocity is a vector because it indicates magnitude and direction.

The motion of the car could also be described as a scalar by saying the speed is 110km per hour and ignoring the direction. Speed is a scalar because it consists of only a magnitude.

Force
An applied force is a vector quantity having both a magnitude and a direction.

Consider a hot air balloon suspended over a farmer's field at a constant altitude. The bouyant force on the ballon pushes the balloon up. At the same time, gravity exerts a force on the balloon pulling the balloon down.

The bouyant force and the force due to gravity act in opposite directions. If they have the same magnitude, the balloon will remain suspended at the same altitude. If the bouyant force is greater than the force due to gravity, the balloon will rise. If the force due to gravity is greater than the bouyant force, than the balloon will fall.

Vector Representation (Two Dimensions)
Forces, and any other vector, may be represented in a number of ways.

Graphically
A vector may be represented graphically by an arrow. The magnitude of the vector corresponds to the length of the arrow, and the direction of the vector corresponds to the angle between the arrow and a coordinate axis. The sense of the direction is indicated by the arrowhead. But when a vector changes direction the unit will change 10x

Polar Notation
In polar notation, the vector is represented by the magnitude of the vector, $$r$$ and its angle from the coordinate axis, $$\theta$$ in the form:

$$\vec A=r\angle\theta$$

or

$$\vec A=(r, \theta)$$

Component Notation
In component notation, a vector is represented by the magnitude of the components of the vector along the coordinate axes.

Scalar Notation
The components of the vector are represented as scalar values which are positive if their sense is in the same direction as the coordinate axis, and negative if their sense is opposite of the coordinate axis. For a vector with a positive x-component and a negative y-component:

$$\vec A = A_x + (-A_y) = A_x - A_y$$

Cartesian Notation
The components of the vector are represented as positive scalar values multiplied by cartesian unit vectors. Cartesian unit vectors are vectors with a magnitude of one that represent the direction of the coordinate axes. The unit vector $$\hat{i}$$ represents the x-axis, and the unit vector $$\hat{j}$$ represents the y-axis. The vectors sense is indicated by the sign of the unit vector. For a vector with a positive x-component and a negative y-component:

$$\vec A = A_x\hat{i} + A_y(-\hat{j}) = A_x\hat{i} - A_y\hat{j}$$

or

$$\vec A = \langle A_x, -A_y\rangle$$

Forces as Vectors
In engineering statics, we often convert forces into to component notation. Replacing a force with its components makes it easier to compute the resultant of a group of forces acting on a body. Conversion of a force from polar to component notation is accomplished by the following transformations:

For force $$\vec F = F\angle\alpha = F_x + F_y$$

$$F_x \ = F \cos \alpha$$

$$F_y \ = F \sin \alpha$$

$$F \ = \sqrt{F_x^2+F_y^2}$$

$$\alpha \ = \arctan \frac{F_y}{F_x}$$

Example 1


Consider a force, $$\vec F$$, with a magnitude, $$ F$$, of 100 N acting in the x-y plane. This force acts at an angle, $$\alpha$$, of 30 degrees to x-axis. What is this force in component notation?

We can replace this force with a pair of forces acting along the x-axis and the y-axis as follows.

$$F_x = F \cos \alpha = 100 cos(30) \ = 86.6 N$$

$$F_y = F \sin \alpha = 100 sin(30) \ = 50.0 N$$

Example 2
Two forces acting in the x-y plane are acting on a point. The first force is 100 N at an angle of 0 degrees. The second force is 50 N acting at an angle of 60 degrees. What is the resultant?

First, resolve the forces into their x and y components. $$ F_{1x} = F_1 \cos (A) \ = 100 cos(0) = 100 $$ $$ F_{1y} = F_1 \sin (A) \ = 100 sin(0) = 0$$ $$ F_{2x} = F_2 \cos (B) \ = 50 cos(60) = 25 $$ $$ F_{2y} = F_2 \sin (B) \ = 50 sin(60) = 43.3$$

Sum all the forces in the x-direction. $$ F_x = F_{1x} + F_{2x} \ = 100 + 25 = 125 $$

Sum all the forces in the y-direction. $$ F_y = F_{1y} + F_{2y} \ = 0 + 43.3 = 43.3 $$

Finally, convert the resultant components back into polar notation. $$ \sqrt{ \left ( \vert F_x \vert ^2 + \vert F_y \vert ^2 \right )} = 132.3 $$ $$ \theta\ = \arctan \left ( \frac{F_y}{F_x} \right ) = 19.1^\circ $$

The resultant force is 132.3 N at an angle of 19.1 degrees.

coming soon!

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