Statics

The word body is a generic yet useful term. We use this word to refer to a solid, physical object; an assembly, system, or structure of multiple solid objects; or even a particle of solid material that lies within a physical object.

In the flipbook, a cat is used as the body.

The cat exerts a force against the platform to launch itself into the air. From a mechanics perspective, we say that the body (the cat) translates (moves) horizontally and vertically.

While this is clearly a dynamics problem, the primary purpose of this flipbook is to:

provide an example for the word body
illustrate the notion of translation

❏ Flipbook: Translation of a leaping cat

Use the arrows to navigate all 5 slides of the flipbook.

The generic term body is sometimes specialized as follows:

the term rigid body is used when we wish to neglect the way a solid material changes shape
the term deformable body is used when we wish to study the way a solid material changes shape

Sometimes, people say that Statics is the study of rigid bodies. This is mostly true, although Lesson 9 - Cables is an exception. We generally won't consider how bodies change shape (or deform) in this course.

🟦 1.2 Types of forces

Forces may be classified as contact forces (that require physical contact) and non-contact forces (also called body forces).

Inspect the chart carefully. You were likely introduced to most of the concepts in your Physics course/s.

Each of these types of forces is explained in further detail in this lesson.

🟦 1.3 The parts of a force vector

In Statics, we use vectors frequently. Force vectors are the most common type of vector we'll use.

It's critical to draw vectors accurately and carefully in Statics, and the level of precision needed in Statics is far higher than what was (likely) expected in your prior Physics course/s.

Be intentional and precise in the way you draw vectors, thinking through each of the four attributes:

the line of action (generally not drawn)
the magnitude (and units)
the direction (arrowhead)
the point of application (head or tail) if the force is a contact force

❏ The four attributes of a force vector

🟦 1.4 Gravitational force (body force)

In some Statics problems, the self-weight of the body is included in our analysis. In other problems, the self-weight of the body is negligible, meaning that we choose to neglect it from our analytical model.

Say that we are studying a mug of coffee.

When we want to account for the weight of the body in our model, we first draw the body (in 2D or in 3D, as shown). Then, we draw a solid dot at the centroid (center) of weight.

For gravitational force (body force), it is typical to align the tail of the vector to the solid dot.

We will learn to calculate the precise location of the centroid later in Statics. For now, simply ensure that the solid dot visually appears near the center of the solid object.

The body force of an object may be represented as W (for weight).

❏ How to draw a body force in 2D and 3D

In Statics, we typically don't work with mass. You still need to know how to convert a mass (m) into a force (or weight, W).

All you have to do is to multiply by the gravitational constant (g):

W = mg

In S.I. units, the gravitational constant is expressed as g = 9.81 m / s².

In U.S. Customary units, the gravitational constant is expressed as g = 32.2 feet / s².

🟦 1.5 Normal forces between solid bodies in contact

❏ Flat surfaces

Two bodies in contact have the ability to transfer a compressive (push) force.

This force is always oriented perpendicular to the contact surface (the interface between the two bodies).

For example, let's revisit the coffee mug. It sits on top of a table. There are two solid bodies (mug and table) in contact.

A normal (perpendicular) and compressive (push) force is transferred at the surface between the mug and table.

We will use the symbol N, which stands for normal force.

❏ How to draw a normal force in 2D and 3D

Important: we only draw a normal force when it is exposed. How do we expose it? We have to remove one of the two solid objects at the interface and replace it with the force.

For instance, in this image, our focus is the experience of the table. Let's personify it a bit: what does the table feel? We do not draw the coffee mug; instead, we show the effect of the coffee mug by replacing its presence with the normal force vector, N.

Let's talk through all four attributes of this vector:

the line of action is the global z-axis
the magnitude of the force is the weight of the mug and contents
the direction is downwards (negative z-direction)
the point of application is the center of the contact area between the two bodies

❏ Inclined and curved surfaces

We must also be able to accurately draw normal forces when the interface between the two solids is inclined or curved.

The trick here is to ensure that the normal force is drawn perpendicular to the surface removed from the drawing.

For instance, in this flipbook, a phone leans against a wall, so that someone can binge their favorite Statics videos while washing dishes. Visualize the phone leaning on the kitchen counter and then work through the flipbook.

We adopt the viewpoint of the phone, taking its perspective and asking ourselves "what does the phone feel?".

From the phone's perspective, it feels a rightward push from the wall at A. It also feels an upwards force from the counter at B.

If you're wondering about friction forces, that's awesome. You indeed need friction if you want the phone to remain in this position. That's discussed in the next section.

❏ Flipbook: the leaning phone

🟦 1.6 Shear (and friction) forces

❏ Overview of shear (and friction)

Let's return to the coffee mug example, and give it a slight horizontal nudge with our hand.

Let's say that there is sufficient friction force between the bottom of the coffee mug and the top of the table to impede motion.

The circular interface between the coffee mug and table transfers a force that is co-planar (or parallel, or in-plane).

For this reason, this type of force is called a shear force. While normal force is perpendicular to a surface; shear force is parallel (or co-planar) to a surface.

The symbol for shear force is V.

Friction force is a subcategory of shear force.

There isn't much consensus on which symbol for friction is best (Ffr or Ff or Fs), so Ffriction is shown in the image.

❏ How to draw shear / friction (in 2D and 3D)

❏ A notational note: how to draw the half arrowhead

In the image, inspect how the 2D friction vector is drawn. It has a half arrowhead (⇁). In the 3D image, it has a standard arrowhead (→). Why does the notation change?

In the 2D view, if we were to draw the normal vector in the proper plane, it would be hard to see, because it would overlap the line that represents the bottom of the mug.

In order to make the drawing more legible, it is customary to slightly offset the shear (or friction) force from its true location. Offset it away from the body slightly. The half arrowhead communicates that we have moved the vector off of its true line of action for clarity. That way, there's no confusion about the true line of action of the vector.

When drawing half arrowheads, make sure that you're always offseting the vector away from the body. In this example, it's below the surface (not above).

🟦 1.7 Internal tension force in a taut cable

Internal forces travel within the fibers of a solid material. They transfer force from one particle to its neighboring particle.

We can only depict these types of forces by making a cut through the solid material.

Let's revisit the coffee mug. Having finished drinking the coffee, we wish to suspend it from a yellow rope. We tie a knot to the handle and let go of the mug.

In order to depict the tension force exerted by the rope on the mug, we must draw a cut plane through it (noted a-a).

In the early part of the course, we will only cut through taut cables (ropes, strings, dental floss, etc.). These materials only have the ability to transfer pure tension (a pull force). That tension force will always align with the geometry of the cable itself.

We generally use the symbol T for tension force in a taut cable.

❏ How to draw internal tension

🟦 1.8 Thinking of force as a push or a pull

So far, we have learned that two bodies in contact have the ability to transfer a push force, and that a cable has the capacity to transfer a pull force.

What about surfaces that are glued together? They can transfer either a push or a pull.

In Physics, you were taught that a force is the action associated with a mass that is accelerated (as defined in Newton's Second Law). That type of thinking is useful for studying bodies in motion, but it's not particularly useful for Statics. It is more useful to think of a force as a push or a pull.

How can you tell a push from a pull? And how do you draw these correctly in Statics? It's all about the vector's point of application.

When the arrowhead of the vector is directed towards the body, it's a push. The head of the arrow is in contact with the body.

The term push is informal (and can be used when working through Statics problems), while compressive force is more formal (and would be the preferred term in an academic journal).

Conversely, when the arrowhead is directed away from the body, we can call it a pull. The arrow tail is in contact with the body.

The term pull is informal (and can be used when working through Statics problems), while tensile force is more formal (and would be the preferred term in an academic journal).

Of course, not all forces are categorized as pushes or pulls:

(1) Body forces, such as self-weight (W = mg), are neither pushes nor pulls.

(2) Friction forces and shear forces are neither pushes nor pulls - they tend to make two parallel planes slip past each other.

❏ The PUSH force

❏ The PULL force

🟦 1.9 Newton's Third Law (N3L)

❏ Overview

You are already familiar with Newton's Laws of Motion.

They were published in 1686 in Philosophia Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), commonly called Newton's Principia.

In Statics, we will be using Newton's Third Law frequently.

The First and Second Laws are not used in Statics; they become important in Dynamics.

We will use the abbreviation N3L to refer to Newton's Third Law.

N3L is commonly paraphrased as "every action has an equal and opposite reaction," but I find Newton's original explanation more directly related to the way we use it in Statics: "the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."

❏ Newton's explanation

From the 1894 English language translation of Newton's Principia by Andrew Motte

❏ How to apply Newton's Third Law (N3L)

Statics students often struggle with correctly applying the Third Law. It's harder than you realize! Here is how to think through it:

Whenever a push (compressive) force is transferred between two bodies, body A pushes on body B, and body B pushes on body A. The arrowheads point towards both bodies (and the point of application is the head of the vector).
Whenever a pull (tensile) force is transferred between two bodies, body A pulls on body B, and body B pulls on body A. The arrowheads point away from both bodies (and the point of application is the tail of the vector).

These forces (called Third Law Pairs) are equal in magnitude and opposite in direction. They have the same point of application and share the same line of action. The example in the flipbook is designed to help you understand this.

❏ Flipbook: simple example of N3L

🟦 1.10 Third Law Pairs in practice

In this flipbook, we will investigate N3L in further detail.

Let's pretend that are applying forces to a pair of wire cutters with our hand, with the intent to cut a wire.

Immediately before cutting through the wire, we take a Statics "snapshot" or freeze-frame. It's like taking a remote control and pausing, to capture and inspect one instant in time.

Work through the flipbook to learn more and see Newton's Third Law in action.

❏ Flipbook: Wire Cutters (Newton's Third Law in Action!)

🟦 1.11 A force is also a tendency to translate

We already know that we can think about forces in terms of pushes or pulls (compressive forces or tensile forces).

Let's flex our thinking a bit: we can also think of force as a "tendency to translate."

In mechanics, the word translate has a very specific meaning. It refers to movement along a specific position vector (or, in a specific direction). For instance, we could talk about an x-direction translation, a y-direction translation, or a translation in any arbitrary direction.

On the checkerboard, there are two checkers. Each is subjected to the same force. The top checker is in translational motion while the bottom checker is in static equilibrium.

The banana provides a reaction that is equal and opposite to the applied force. Since the tendency to translate is arrested by the banana, we know that there is sufficient friction below the banana to impede motion.

❏ Animation: A force is a tendency to translate

The top checker is in motion and the bottom checker is in static equilibrium.

While we do not study motion in Statics, it is still useful to think of a force as a tendency to translate. In other words, for the bottom checker, the applied force tends to cause translation, even though the banana prevents the motion.

Such thinking allows us to take an enormous intellectual leap forward. If we were to remove the banana entirely, then the checker would translate rightward. Therefore, in order to keep the checker in static equilibrium (prevent motion), we can deduce that the banana's reaction force must be leftward (←).

🟦 1.12 Notational differences from your Physics course

You'll need to "unlearn" some habits that you were likely taught in your Physics course. Engineers use a slightly different language than Physicists, and that's OK. It's like Spanish vs. Portuguese - they are similar ways to communicate, but certainly not identical.

In Physics, forces generally were notated as an F with a subscript. This is because in Physics, other vectors (velocity, acceleration, etc.) are also shown.

In Statics, we don't study velocity or acceleration, so we tend avoid using the letter F as a symbol for force. Instead, we prefer the more specialized symbols introduced in this lesson:

You may have also noticed that in the Statics column, the symbols aren't wearing any arrows as hats. Are they still vectors? Yes, absolutely! It's just that when we are doing 2D Statics problems, we typically don't include the arrows in our notation. The fact that they are vectors is implied by context.

🟦 1.13 Summary of symbols and units for force

❏ Summary of symbols for force

We would like to communicate effectively by using symbols for the different types of forces we will encounter in Statics. Please use these symbols in this class (avoid using the symbols you used in Physics):

F = a common, generic symbol for force (F1, F2, etc. for multiple forces)

FR = a common symbol used for a resultant force (e.g. the sum of two vectors creates a resultant force)

Fx = used to indicate the x-direction component of a force (projection on the x-axis)

Fy = used to indicate the y-direction component of a force (projection on the y-axis)

P = a common, generic symbol for force (the P stands for "point load"); use P1, P2, etc. for multiple forces

W = the body force or the weight of the body, always drawn at the center of weight (or centroid)

N = a normal, compressive force (perpendicular to the contact surface between two bodies) -- always a push

T = an internal tensile force; always draw a cut plane to expose internal tension force

V = a shear force (one that is parallel to a plane or that lies within a plane)

Ffr = the force of friction (a type of shear force), always located in the plane of the surface between bodies

Fs = the force in a (translational) spring, which could be either compressive or tensile

The diagrams below provide examples of the different symbols we might use to show different types of forces on a wheelbarrow.

Several bodies interact with a wheelbarrow full of soil.

Here are the effects of the bodies interacting with the wheelbarrow.

❏ Units of force

Sometimes, we will express units of force in U.S. Customary Units. Other times, we will use S.I. (or metric) units. Always work problems in what ever measuring system is given to you. If you are given a problem in pounds, and you answer in Newtons, it's like answering "tutto a posto" (everything's fine, in Italian) when someone asks "hoe gaat het" (How's it going, in Dutch). Technically, it's the correct answer, but it's not a useful way to communicate.

In U.S. Customary Units, the base unit is the pound-force, or pound. Be sure not to confuse it with the pound-mass.

For this class, the preferred symbol for the pound (force) is the hashtag, or # symbol.

Some people (especially outside the U.S.) prefer to abbreviate pound as lb. or lbs. in their engineering calculations. Others use lbf. It's confusing, I know. I'm sorry.

A hammer weighs approximately 1#. (Public domain)

There is another important unit of force to know in U.S. Customary Units: the kip or kilopound.

As you may have guessed, 1 kip is equal to 1,000 pounds (1,000# or 1E3#).

We say "kip" when speaking, but generally abbreviate as k in written calculations.

An adult horse weighs around 1 kilopound.(Image credit: Srdjan Živulovič)

You're already familiar with S.I. force units from Physics. We will use Newtons (N) for small scale problems.

For larger scale problems, we will typically use kilonewtons (1 kN = 1 E3 N).

Occasionally, we even have to use meganewtons (1 MN = 1 E6 N = 1 E3 kN).

An apple weighs around 1N. (Image credit Fir0002/Flagstaffotos)

🟦 1.14 Components, resultants, and vector addition

❏ Components for a vector specified by an angle

We often will want to use vector operations to simplify force vectors.

You can break an inclined vector down into its x-direction and y-direction components (for any xy coordinate system).

Consider a force vector that is inclined at angle θ from the positive x-axis.

The effect of this force can be represented by its two components, in combination.

Sketch a bounding box around the vector and construct the two components as shown.

Super duper important: components are always drawn tail-to-tail or head-to-head. They are never drawn head-to-tail or tail-to-head. This is a very common error!

The force vector can be expressed in 2D vector notation in two ways:

F = Fx i + Fy j ←more common in Physics courses

F = <Fx, Fy> ←more common in Engineering courses

How to use trigonometric identities to write expressions for the components of the vector

❏ Components for a vector specified by the rise over the run

A more practical way that engineers like to specify the inclination of a vector is by specifying the rise (y) over the run (x).

We still want to break this inclined vector into its components, but the process is different.

You don't want to waste time calculating the angle of inclination with θ = tan-1 (x/y). A better approach is to use ratios to solve for the component forces:

R = (x2 + y2)1/2

Fx = (x/R)F

Fy = (y/R)F

How to use ratios to write expressions for the components of the vector

❏ Vector addition

You can also add a system of vectors together (using the head-to-tail method you learned in prior classes) if it's advantageous to create a single resultant vector.

For vector addition, use the head-to-tail logic you learned in prior studies.

How to simplify systems of forces into a single resultant force

❏ In summary

Components are always head-to-head (for a push force) or tail-to-tail (for a pull force).

The only time you will ever use head-to-tail logic is when you are adding vectors together.

🟦 1.15 Parenthetical notation vs. chevron notation

❏ Parenthetical notation

We use parentheses to designate the location of points.

For example, (0, 0) cm designates the origin (O).

Be sure to include units after the parentheses.

Use parentheses for ordered pairs

❏ Chevron notation

We use chevrons to designate vectors.

Points are absolute, but vectors are relative. F1 is applied at both A and O.

Again, don't forget to include units with your vectors.

Use chevrons for vector components

❏ Notation for resultant forces

You can use single or double vertical lines to indicate that you're calculating a resultant force, like this:

| F | = (Fx2 + Fy2)1/2

|| F || = (Fx2 + Fy2)1/2

You can draw resultant forces like this

➜ Practice Problems

Notes:

(1) The only way to master Statics (or any other engineering topic) is to work practice problems. Approach each new problem with curiosity and an engineering mindset ("I can figure this out!"). It's not easy to learn engineering, but you can do it, as long as you put in the work.

(2) This problem set requires use of concepts in prerequisite coursework (Physics and Mathematics).

(3) I have built this site on a platform that does not provide an easy way to write subscripts. If you see F1 or F_1, please interpret it as F₁ . I'm in the process of fixing this, but it will take some time! Thanks for your patience.

Problem 1

A force, F1, is known to have an x-direction component of +80N and a y-direction component of -50N.

Make a sketch of F1
Write F1 in vector notation
Solve for the magnitude of F1.

Image is intentionally excluded; the recipe for the geometry is in the problem statement.

Problem 2

A force, F2, aligns with the x' axis. (In this instance, the prime symbol just indicates a different Cartesian coordinate system; it does not signify a derivative).

Draw the image yourself (wait, isn't this a waste of time? can't I just look at the picture? It's not a waste of time because the act of re-drawing the figure is a way to activate your brain.)
Sketch a bounding box around F2
Solve for the components of F2 in the xy coordinate system in terms of the angle alpha (α).
Now, solve for the components of F2 in the xy coordinate system in terms of the angle beta (β).

Problem 3

You and some friends are trying to push a car backwards up a ramp.

Let's say that all of your combined effort (pushing, friction, multiple hands in multiple locations) is equivalent to the force F3 depicted. For the purpose of this problem, don't be concerned about whether the car is in motion or not; maybe the force is enough to overcome friction, and maybe it's not.

Sketch the problem geometry (simplify it and distill it to the basics, meaning that you don't have to draw the car itself)
What is the component of F3 that is parallel (or planar, or in-plane) with ramp surface a-a? Express your answer numerically (with a decimal, not a sine or cosine).

Problem 4

A cuboid (a box that isn't a cube) is defined by a position vector r1 = <3,3,5> feet that originates at (0,0,0) feet. That is, the tail of the vector is located at (0,0,0) feet.

Force F4 is applied to the cuboid at Point A. The vector's tail lies at coordinates of (1,2,0) feet. F4 = <-2,4,-3> kips.

Sketch the basic scenario of this problem. If hand drawing is a challenge area for you, it is perfectly fine to use a CAD tool (for those of you that have a skillset in 3D modeling).
Is F4 a push or a pull?
What is the magnitude of F4?
What can you infer from this problem statement about the proper use of parenthetical notation like (1,2,3) vs. chevron notation like <1,2,3>?
Did you notice how I used units with both parenthetical notation and chevron notation? What's your strategy to remember to include units with your vectors when working Statics problems?

Image is intentionally excluded; the recipe for the geometry is in the problem statement.

Problem 5

Force F5 has a line of action that lies along position vector <-3,4,0> meters.

You may sketch this if needed, or visualize in your head if you prefer.
If the magnitude of F5 is 10 kN, what is the x-component of F5?
Explain how you can solve this problem very quickly, and without the use of a calculator.

Problem 6

Force F6 has a line of action that is parallel to the hypotenuse of right triangle ABC.

If you know that F6x has a magnitude of 8MN, what is the magnitude of F6y?
Did you notice that I didn't draw the x and y axes in the figure? When they are omitted from the sketch, a reasonable person will assume that the author intended positive x to be rightwards and positive y to be upwards. This is true in my class; please note that some other professors want axes on every single drawing no matter what.

Please note that the use of a little floating triangle next to an angled or inclined line is a very common way for engineers to communicate the aspect ratio of the rise and run. In real-world engineering, the use of ratios to specify angles is just as common as using degrees (and in engineering practice, no one would use radians).

Problem 7

This problem has 4 parts:

Draw the 45 degree right triangle. Dimension the length of the hypotenuse as 1, and dimension the two legs as root 2 over 2.
Memorize that the sine of 45 degrees is equal to the cosine of 45 degrees is equal to root 2 over 2.
Draw another 45 degree right triangle. Dimension the length of the hypotenuse as 1, and dimension the 2 legs as the decimal equivalents of root 2 over 2 to 4 significant figures.
Memorize the decimal equivalent of root 2 over 2.

Problem 8

This problem has 4 parts:

Draw a 30-60 right triangle. Dimension the length of the hypotenuse as 1, and dimension the two legs as 1/2 and root 3 over 2.
Memorize the sine and cosine of 30 degrees (and 60 degrees).
Draw another 30-60 right triangle. Dimension the length of the hypotenuse as 1, and dimension the two legs with the appropriate decimal equivalents (to 4 sig. figs.).
Memorize these two decimal equivalents of the sine and cosine of 30 degrees (and 60 degrees).

Problem 9

A right triangle has a hypotenuse of 10m as shown.

Let's say that you know that the sine of alpha (α) is precisely 0.400.

Based on your memorization work, above -- is alpha greater than or less than 30 degrees?
Without drawing the figure, and without using a calculator, solve for length d.

Problem 10

Angle gamma (γ) has a known cosine of 6/7.

Based on your memorization work, above - is gamma greater or less than 30 degrees?
If length AB is 21 cm, what is the length of line AC?
Can you deduce that the sine of angle gamma is equal to 1/7? Why or why not?

Problem 11

A heavy box (weight of W) lies on a ramp (or an inclined plane). The ramp a-a is parallel to the hypotenuse of right triangle ABC.

You need to convert the weight, W, into components in the x' and y' direction.

Draw the key parts of the image. Make an effort to duplicate the 2:3 ratio in your graphic. (You can estimate this, with ratios, or measure it.)
Sketch in a bounding box around W in the x'-y' coordinate system.
Write the components of W with respect to the x' and y' directions in vector notation and also in terms of root 13.