🟦 1.1 Translation
In this class, we will often speak of "bodies" or "rigid bodies."
This is a very generic term. We use it to refer to a solid, physical object (or an assembly of such objects).
Here we will use a cat as the body.
Due to the force between the cat's feet and the platform, the cat is launched into the air!
From a mechanics perspective, we say that the cat translates (moves) horizontally and vertically. The position of the cat is a function of time.
Browse the images in the flipbook to refresh your memory on a few important concepts you learned in Physics, including the free-body diagram (FBD).
❏ Flipbook: Translation of a leaping cat
🟦 1.2 Newton's Laws
In Physics, you learned about Newton's Laws of Motion. They were published in 1686 in Philosophia Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). This book is commonly called Newton's Principia.
Here is a quick refresher from Physics:
❏ Newton's First Law
Unless an external force is applied:
a body in constant motion will remain in constant motion
a body in static equilibrium will remain in static equilibrium
❏ Newton's Second Law
Force equals mass times acceleration.
❏ Newton's Third Law
Whenever a "push" force is transferred between two bodies, body A pushes on body B, and body B pushes on body A.
Whenever a "pull" force is transferred between two bodies, body A pulls on body B, and body B pulls on body A.
These forces (called "Newton's Third Law Pairs") are equal in magnitude and opposite in direction.
🟦 1.3 A push or a pull
Force can be defined in a few ways.
For instance, you can think of a force as the action associated with a mass that is accelerated (as defined in Newton's Second Law). That type of thinking is very useful when you are studying bodies in motion. But that framework isn't useful for Statics.
In Statics, it's more useful to think of a force as either a push or a pull between two bodies in contact.
When the arrowhead of the vector is directed towards the body, we call it a push. That's an informal term, though. If wanted to be more formal, we could call it a compressive force (or, one that transfers compression).
Conversely, when the arrowhead is directed away from the body, we can call it a pull (informally). The formal term for this idea is a tensile force (or, one that transfers tension).
Be careful: not all forces are categorized as pushes or pulls. The self-weight of a body is indeed a force, but it is neither a push nor a pull. Friction forces and shear forces are neither pushes nor pulls - they are forces that tend to make two parallel planes slip past each other.
❏ Key concept: a push vs. a pull
🟦 1.4 Third Law Pairs
In Statics, we will use Newton's Third Law with great frequency. That's the one that you may have heard expressed as "every force as an equal and opposite reaction." It's better to think of it in terms of forces that are equal in magnitude but opposite in direction when two bodies come into contact.
Let's pretend that are applying forces to a pair of wire cutters with our hand, with the intent to cut a wire. Before the wire is cut, we take a Statics snapshot, and analyze the way forces move through the system. Work through the flipbook to learn more and see Newton's Third Law in action.
❏ Flipbook: Wire Cutters (Newton's Third Law in Action!)
🟦 1.5 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 x-direction translation or y-direction translation. Or we could also investigation the translation along any unit vector, such as the unit vector <3/5i, 4/5j>.
On the checkerboard, there are two checkers. Each is subjected to the same force. The top checker is in motion (specifically translational motion) because there is no static equilibrium.
❏ Animation: A force is a tendency to translate
The bottom checker is in static equilibrium. The banana provides a reaction that is equal and opposite to the applied force. (Note that if the force was large, then we could overcome the friction between the banana and chessboard, and both the checker and banana would be in motion.)
While all Statics problems are in static equilibrium by definition, it is incredibly useful to think of a force as a tendency to translate. In other words, for the bottom checker, the applied force tends to make the checker translate to the right. If we were to remove the banana, the checker would tend to move to the right. Therefore, in order to keep the checker in static equilibrium, the banana must react with a leftward-acting reaction force.
🟦 1.6 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).
🟦 1.7 Normal compressive forces between bodies in contact
🟦 1.8 A force distributed over an area or a length
In Physics, you represented force as a single vector. The implication of modeling force in that way is that the force can be applied at a finite point, such as the tip of a pushpin that is unimaginably sharp.
In Statics, we will sometimes distribute a force over an area.
Sometimes this may be an average distribution. Other times, we may need a higher order function (perhaps linear, or even nonlinear) to approximate the real world.
This idea -- force per area -- is given the term area load. Because the units (force per area) are the same as they are for pressure, we will use the symbol p for area loads.
Pushing on a pushpin
Sometimes, it is also useful to take a force, and divide by a length. The result is called a line load.
We will use the symbol w for line load.
You can also think of this as "force per distance" or "force intensity."
We will often be converting line loads to equivalent forces by taking the area under the curve.
NOTE! In a 2D projection, a line load and an area load look identical. The only way to tell them apart is by inspecting the units. Line loads are force per distance and area loads are force per area.
Converting a concentrated force (or point load) to a force per distance (or line load)
🟦 1.9 Components and resultants of a force vector
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
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
Finally, work problems 1 and 2 at the Mechanics Map website:
http://mechanicsmap.psu.edu/websites/A1_vector_math/A1-2_vectoraddition/vectoraddition.html