What is mechanical equilibrium? Why do we study it?
This introduction comes from Being Brunel: Notes From a Civil Engineer
If civil engineering was religion (and in a way it is; institutionalised by men in funny hats), the first commandment would be: “Thou shalt always have static equilibrium”
The principle is easy: the sum of all the actions acting on a structure should come to zero.
Before Newton invented equilibrium in 1686, the world remained in check because of Aristotle‘s assertion that everything has its natural place. That is: rocks like to be on the ground, and therefore they fall, unless propelled by an external agent.
About 1800 years later Gallilio chipped-in; removing the sense of belonging that came from a ‘natural place’ and instead suggesting that things move at a constant speed (which includes ‘at-rest’, unless you’re at the gym) until they are forced to change.
With his best seller: the Philosophiæ Naturalis Principia Mathematica Newton refined these theories with over 200 experiments (which seems overkill to me) and derived the infamous laws of motion. As far as Civil Engineers are concerned this was when it all began; hence the BN/AN (Before-/After- Newton) calendar we use amongst ourselves. The logic goes as such:
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Structures neither move, nor do they accelerate (this is an axiom that the public insists on…)
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Therefore the ‘velocity’ of the structure, and all of the components that make it, is always zero.
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That means that the net force on the structure is zero (Newton’s first law)
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All forces applied to a structure are due to accelerating masses (Newton’s second law)
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At a global level these actions on the structure are resisted by the reactions of its supports (Newton’s third law)
Like all good articles of faith, however, this doesn’t quite stand up to scrutiny; that is- Newton took the assumption that everything is a single point of mass.
50 years later, however, Euler wrote a second testament: “Laws of Motion 2: Motion Harder“, 🤣 which bridges the gap. In practice, however, the distinction is rarely made by Civil Engineers.
So the next time you pick-up some engineering calculations (as I’m sure you do everyday), look-out for the incantation ΣF = 0. By invoking this statement, engineers call the structure into equilibrium; and once you know that a body is in equilibrium, working out the forces at any point within the structure is as simple as balancing the equation.
Mechanical Engineering PowerPoint slides
from Conceptual Physics, Paul Hewitt, Chapter 2
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At this point we note that we use algebra in physics. Algebra uses letters as variables.
There are so many different things to measure in science that even the 26 letters of the English alphabet are not enough. Therefore we have to use the same letter to stand for different things.
a – absorbance (chemistry,) acceleration, ampere, area, chemical affinity
b – magnetic field, molality, impact parameter (in nuclear physics)
c – capacitance, speed of light, molar heat capacity, coulomb
d – diameter, distance, dose (radioactivity,) Debye (electrical dipole moment,) Diffusion coefficient
s – seconds, arclength, entropy, Sulfur, Siemens (unit of electric conductance)
With the thousands of possible things to measure (and in need of an abbreviation) people thought that it would be better if we have more than just 26 letters to choose from.
We get these additional letters from the Greek alphabet. That’s important today we use a Greek letter to abbreviate the word “sum”.
Sum is abbreviated with the Greek letter Sigma (∑) .
Here is how we mathematically describe an object in a state of equilibrium.
We say that the sum of all forces on it adds up to zero.
Here is an example of a worker on a skyscraper window cleaning apparatus. This is a type of Bosun’s chair, extended into a platform.
Hopefully the sum of all forces on it will add up to zero – otherwise it is accelerating, rotating, or falling!
We may signify it like this:
Here the sum of forces = 0, so the scaffold is in equilibrium.
The “normal” force
What forces act on a book, lying at rest on a table?
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Weight = the force of gravity pulling the book’s mass down
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The normal force, from the table, pushing back up on the book
Doesn’t seem like there would be two forces, does there? But if there was only one force, then the book would be accelerating down – it would be picking up speed, and moving through the surface of the table. Yet that obviously doesn’t happen.
Try it yourself – push your own hand down on your desktop. It won’t go through. You’re pushing down…but do you feel the table pushing back up on your hand? That’s the normal force that you are feeling!
Where does the normal force really come from?
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Your hands are made of atoms, and so is the desk
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Atoms have protons (+) on the inside, and electrons (-) on the outside
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When you push your hands down on the desk, your hand’s atoms push against the desktop’s atoms.
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But only the outer electrons of your hand’s atoms, come near the outer electrons of the desktop’s atoms.
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And all electrons repel each other. Remember: like charges repel (just like opposite charges attract)
The upward (normal) force = +
The downward force = –
They cancel out, so the net force is 0. No net force means no acceleration!
An object at rest is one form of equilibrium. Static equilibrium.
An object moving at constant speed, in a straight line, is another example of equilibrium. Dynamic equilibrium.
So what about equilibrium for moving objects? Once in motion, if there is no net force to change the motion, then the object stays in motion!
Friction is a contact force between two objects. If the net force = 0, then that means the Push force is equal to (but opposite in direction to) the friction force.
Adding vectors
Vector addition from the PhysicsClassroom
Parallelogram Method of Vector Resolution
Finding the Resultant when adding vectors
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An equal yet opposite vector is needed for equilibrium – notice the dashed line vector.
Using the parallelogram rule we find that the tension in each rope is more than half of her weight.
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Labs
Mechanical equilibrium lab
Learning Standards
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion is a
mathematical model describing change in motion (the acceleration) of objects when
acted on by a net force.
HS-PS2-10(MA). Use free-body force diagrams, algebraic expressions, and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations
A FRAMEWORK FOR K-12 SCIENCE EDUCATION: Practices, Crosscutting Concepts, and Core Ideas
PS2.A: FORCES AND MOTION
How can one predict an object’s continued motion, changes in motion, or stability?
Interactions of an object with another object can be explained and predicted using the concept of forces, which can cause a change in motion of one or both of the interacting objects… At the macroscale, the motion of an object subject to forces is governed by Newton’s second law of motion… An understanding of the forces between objects is important for describing how their motions change, as well as for predicting stability or instability in systems at any scale.
Massachusetts Science and Technology/Engineering Curriculum Framework (2006)
1. Motion and Forces. Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.4 Interpret and apply Newton’s three laws of motion.
1.5 Use a free-body force diagram to show forces acting on a system consisting of a pair of
interacting objects. For a diagram with only co-linear forces, determine the net force acting on a system and between the objects.
1.6 Distinguish qualitatively between static and kinetic friction, and describe their effects on the motion of objects.
1.7 Describe Newton’s law of universal gravitation in terms of the attraction between two objects, their masses, and the distance between them.
1.8 Describe conceptually the forces involved in circular motion.
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