Statics

Lesson 01

Force -- or, a tendency to translate

🟦 1.1 Translation of bodies

🟦 1.2 Types of forces

🟦 1.3 The parts of a force vector

🟦 1.4 Gravitational force

🟦 1.5 Normal (compressive) forces between solid bodies in contact

🟦 1.6 Shear (and friction) forces

🟦 1.7 Internal tension force in a taut cable

🟦 1.8 Thinking of force as a push or a pull

🟦 1.9 Newton's Third Law (N3L)

🟦 1.10 Third Law Pairs in practice

🟦 1.11 A force is also a tendency to translate

🟦 1.12 Notation, Symbols, and Units

🟦 1.13 Components, resultants, and the bounding box

➜ Practice Problems

🟦 1.1 Translation of bodies

In Statics, we will often speak of bodies.

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

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 10 - Cables is an exception to this generality. In general, we 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. They are used throughout the course.

It's critical to draw vectors accurately and carefully in Statics.

Be forewarned: the level of precision needed in Statics is far higher than what was (likely) expected in your prior Physics course/s. You will need to be intentional and precise in the way you draw vectors. We can't just sprinkle them in random places on your Statics drawings.

Here are the four attributes of a force vector:

the line of action (generally not drawn)
the magnitude (and units)
the direction (arrowhead)
the point of application (head or tail)

❏ The four attributes of a force vector

🟦 1.4 Gravitational 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 in our mathematical 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 draw the body (in 2D or in 3D, as shown), and then draw a solid dot at the centroid (center) of weight. Align the tail of the weight vector to the solid dot as shown.

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

Recall that you can convert mass (m) to force by multiplying by the gravitational constant (g): W = mg

Here is the gravitational constant in both systems of units:

g = 9.81 m / s² S.I. units

g = 32.2 feet / s² U.S.C. units

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

🟦 1.5 Normal (compressive) 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. Therefore, we have the common scenario of two solid bodies in contact.

A normal (perpendicular) and compressive (push) force is transferred at the interface below the mug and above the table. We can call it N, which stands for normal force.

Important: we can 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
the point of application is the center of the contact area between the two bodies

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

❏ Inclined and curved surfaces

We must also be able to accurately draw normal forces when the interface between the two solids is more complicated. It always takes practice for Statics students to master the skill.

The trick here is to make sure that the force is perpendicular to whichever surface is removed from the drawing.

For instance, in this flipbook, a phone leans against a wall. Perhaps the phone's owner wants to re-watch 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 (again, we can personify one of the solid elements, taking its perspective and asking ourselves "what does the phone feel?").

From the perspective of the phone, the wall is pushing rightwards at A. The counter is pushing upwards at B.

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

❏ Flipbook: the leaning phone

🟦 1.6 Shear (and friction) forces

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. Normal forces are perpendicular to a surface; shear forces are parallel to a surface.

Friction force is a subcategory of shear force.

In the image, note how the 2D friction vector is drawn. We could draw the normal vector in the proper plane, but graphically, it would be hard to see this, as 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. We communicate that we have moved the vector off of its true line of action, by using a half-arrow. The half-arrow head is always on the far side, relative to the planar surface. In this example, it's below the line of the vector.

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

🟦 1.7 Internal tension force in a taut cable

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

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 pass a cut plane through it (noted a-a on the figure).

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

Later in the course, we will cut through more complex solid members, which are capable of developing other types of internal forces.

❏ How to draw internal tension (in 2D)

🟦 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. Surfaces that are glued together 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.

❏ Key concept: the PUSH force

❏ Key concept: the PULL force

🟦 1.9 Newton's Third Law (N3L)

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 rarely (if ever) used in Statics; they are at the core of the study of Dynamics.

We will use the abbreviation N3L to refer to the Third Law.

N3L is commonly paraphrased as "every action has an equal and opposite reaction."

❏ 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. Work through the example problem in the flipbook.

❏ Flipbook: simple example of N3L

🟦 1.10 Third Law Pairs in practice

In this flipbook, we 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 paushing 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 to impede motion.

❏ Animation: A force is a tendency to translate

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-acting (←).

🟦 1.12 Notation, Symbols, and Units

❏ Notation for force

In Physics, your professors likely put little arrows on top of vectors. This was to help you remember that a vector has magnitude and direction. In Statics, we rarely deal with any concepts that aren't vectors. For instance, force is a vector. We typically do not use that little arrow symbol on top of our vectors, because everyone who speaks the language of solid mechanics already knows that force is a vector.

❏ Symbols for force

We would like to communicate effectively by using symbols for the different types of forces we will encounter:

W = the weight of the body, always drawn at the center of weight (or centroid), equal to mass times acceleration (which can be expressed as W = mg)

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

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

FR = a common symbol used for a resultant force

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

T = a tensile force (such as the force in a wire or cable)

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

Ax = a force (usually unknown) at node (or point) A in the x-direction

Ay = a force (usually unknown) at node (or point) A in the y-direction

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

Ffr = the force of friction, always located in the plane of the surface between bodies (a friction force can be thought of as a subcategory of a shear force)

The diagrams below provide examples of the different symbols we might use to construct a free-body diagram of 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 a weird 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. 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. There is another important unit of force to know: the kip or kilopound. As you may have guessed, 1 kip is equal to 1,000 pounds (1,000# or 1 E3 #). The kip is often abbreviated as a single k.

The S.I. Units of force will be familiar to you from Physics. We will use Newtons (N), kilonewtons (1 kN = 1 E3 N), and occasionally meganewtons (1 MN = 1 E6 N = 1 E3 kN).

🟦 1.13 Components, resultants, and the bounding box

Force is a vector, and vectors have magnitude and direction. We often will want to use vector operations to simplify vectors.

For instance, you can add a system of vectors (head-to-tail) if it's advantageous to create a single resultant vector. Sometimes that's helpful and sometimes it's not.

You can also break an inclined vector down into its x-direction and y-direction components (for any xy coordinate system). Again -- sometimes that's helpful, and sometimes it's not. Statics problems require strategic thinking.

Go through this graphic carefully. Especially, please note that vector components must be drawn tail-to-tail or head-to-head. This is different than head-to-tail vector addition.

Erroneously drawing vector components head-to-tail is a VERY common error in Statics. The sooner you learn to avoid this, the better.

➜ Practice Problems

The only way to master Statics (or any other engineering topic) is to work problems. Work as many as possible. Approach each new problem with curiosity and an engineering mindset ("I can figure this out!"). It's not easy to learn this material, but you can do it, as long as you put in the work.This first problem set does have some review concepts from prerequisite coursework; reach out if you need help on these topics.

Note to Fall 2024 students: I'm building this site in Google and there is not an easy way to do subscripts. When I'm first drafting the website, I don't bother to insert the subscripts. If you see something like F1, please interpret it as F₁ . If you are the slightest bit confused about whether or not a symbol is intended to be a subscript, please just ask.

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. We're not 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 (it is 100% OK to 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, which lies at coordinates of (1,2,0) feet. F4 = <-2,4,-3> kips.

Sketch the basic scenario of this problem. If visualization 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). It is also OK to build a rough facsimile (find a cardboard box or fold up paper).
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 3 parts:

Derive the fact that the sine of 45 degrees is equal to the cosine of 45 degrees is equal to root 2 over 2.
Memorize this fact; it will be very useful in your Statics problem-solving practice.
Also memorize the decimal equivalent to 4 significant figures: 0.7071

Problem 8

This problem has 3 parts:

Derive the fact that the sines and cosines of the 30-60 triangle are 1/2 and root 3 over 2.
Memorize this fact; it will be very useful in your Statics problem-solving practice.
Also memorize the decimal equivalent of root 3 over 2 to 4 significant figures: 0.8660

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

Being a creative type of person, you designed some custom playing cards. Your card deck measures 2" x 3". You pull out the ace of spades and lay it on a table. The self-weight of the card is negligible, and we choose to ignore friction in this problem, as both the table and the playing card are both very smooth.

The origin of the Cartesian coordinate system is the center (centroid) of the card and gravity acts in the negative z-direction.

You glue toothpicks to the card at the following points and then apply forces to the card via the toothpicks as shown in the table. (Why are you doing all this? I have no idea. Just go with it.)

Draw a picture that illustrates the card and the forces. These are the effects of the toothpicks, so you do not need to draw the toothpicks in the image; just the forces.
What is the net force in the x-direction?
What is the net force in the y-direction?

Problem 12

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.

Problem 13

Two forces are applied to particle (or point) A:

F1 = <4,4> kN and F2 = <-6,-1> kN.

What is the resultant force associated with F1 + F2? Express this in vector notation.
What are the x-direction and y-direction components of the resultant force? You can simply write the magnitudes, but use a sign to indicate positive or negative direction.

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

Problem 14

Are you confused, feeling like these problems didn't have anything to do with the reading assignment?

You're kind of right - they were mostly review problems from skills you learned in prerequisite coursework. Hopefully, they were mostly review material, with a few new tips and tricks mixed in. All of the content from this reading is CRITICALLY important on a conceptual level, but not well-suited for homework problems.

For this last practice problem, make some quick bulleted notes for yourself that summarize key concepts from the reading. This isn't busy work, it's helping you learn and retain important information.

what's the difference between rigid bodies vs. deformable bodies?
what is the difference between contact forces and non-contact forces?
what is translation? why do we think of a force as a "tendency" to translate?
what is the key difference between a normal force and a shear force?
what are the four parts of a force vector, and how does the point of application define pushes vs. pulls?
fill in the blank: normal forces between bodies are perpendicular with respect to the surface _________ the drawing
fill in the blank: internal forces require a _____ plane
what is the significance of N3L in your own words?
make sure that force components are drawn head-to-head or tail-to-tail, not (fill in the blank) ___________

All concepts from today's reading will be CRITICALLY important over the next few lessons, and throughout the course.

OPTIONAL: Want some additional practice? Feel free to check out these resources at the Mechanics Map website:

problems 1 through 3:

http://mechanicsmap.psu.edu/websites/A1_vector_math/A1-1_vectors/vectors.html

problems 1 and 2:

http://mechanicsmap.psu.edu/websites/A1_vector_math/A1-2_vectoraddition/vectoraddition.html

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