Statics

Lesson 01

Force -- or, a tendency to translate

🟦  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:

❏ Flipbook: Translation of a leaping cat

The generic term body is sometimes specialized as follows:

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 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:

❏ 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 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

❏ 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:

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 and resultants

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.

➜ 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.

Warm-up:

Also, work problems 1 through 3 at the Mechanics Map website:

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