Here you can find information on all aspects of engineering. Our blog covers a wide range of topics, including mechanical, electrical, civil, and chemical engineering, among others. So join us on this exciting journey into the world of engineering, and let CalcTree be your guide!

Engineering

Sustainability

Construction

Physics

Mathematics

Technology

Articles & Resources

Explore a wealth of engineering articles & resources specially designed for students. Stay informed with the latest trends & insights. Read now

Engineering Articles & Resources - CalcTree

Thermodynamics

Logic Gates explained with calculators

Applications of Fourier Series in Signal Processing and Engineering

Fundamentals

Matrix Algebra and its applications in Engineering and Physics

Types of Pulleys & Calculators

Enhance your understanding of physics and its application to practical science and engineering here.

Physics Articles - CalcTree

Energy–momentum relation

Beam Analysis and Design Fundamentals

Fundamentals: Second Moment of Inertia

Fundamentals: Newton’s 3 Laws of Motion

Master the Fundamentals of Mathematics with Calctree. Boost your skills with interactive lessons and step-by-step solutions. Start learning today

Fundamentals of Engineering 

Master Beam Analysis Fundamentals with Calctree. Learn the principles, calculations, and techniques to analyze beams effectively. Read now

Beam Analysis and Design Fundamentals - CalcTree

Whenever you see or want to describe the effect of forces on basically any object, 

 likely be using Sir Isaac Newton's Laws of Motion to do it. These laws are the cornerstone of modern physics. They substantiate everything from our knowledge of planetary motion to our understanding of how planes take off and fly. This article will describe Newton's Laws of Motion and how they apply to everyday life.

An object at rest remains at rest, and an object in motion remains in motion at a constant speed and in a straight line unless acted on by an unbalanced force

: Things like to stay where they are; Newton's first law of motion describes an object's tendency to keep doing what it's doing.

 states that all objects will remain at rest or in uniform motion unless an external force changes its state. Changing how something moves requires acceleration, and you need a net force.

An object's tendency to resist change in a state of motion or stationery is known as 

. How is inertia measured? Well, you need to know about mass. Say you throw two balls; one is a bowling ball, and the other is an inflatable ball. Intuition tells you which is harder to move or throw and harder to stop. The bowling ball has more mass; therefore, it has more inertia…more stuff to keep it doing what it's doing before you stop it.  

Newton's first law also applies to bodies at rest. Say you have an object lying on a table. The force of the object downwards onto the table due to its weight and the equivalent upward reaction exerted by the table means that all external forces cancel each other out, and the object remains at rest. In other words, there is no net force acting on the object. 

The same situation would happen if you applied equal but opposing horizontal forces on the object (provided it was on a smooth surface). Say you put two hands on an object from opposite sides and apply the same amount of pressure on each side, the external forces (from your hands) would cancel each other out, and the object will remain stationary. The object will move when the forces are unbalanced (i.e., you remove one hand). Then do the same thing on a rough surface; it might be harder to slide the object across that surface because you've added another force into the mix this time. It's friction.

The concept of inertia and forces connects to Newton's second law.

Newton's second law of motion states that the acceleration of an object depends on the object's mass and the amount of force applied. 

 To overcome inertia, you must apply an unbalanced or net force to an object. A 

 is all the forces left over after adding together all the forces that might otherwise cancel each other out. When forces get unbalanced, that's when exciting stuff happens!

The net force is found using the equation:

You've probably heard 'force equals mass times acceleration' over and over, and for a good reason. It's likely the most used equation in physics. This equation tells us that an object subjected to a net external force will accelerate. The amount that an object will accelerate is directly proportional to the magnitude of the force. Acceleration is also inversely proportional to the mass of the object. This goes back to our example of two balls, one light, and one heavy. If the same force is applied to both, the heavier ball will experience less acceleration than the lighter ball because it has a higher mass.

 the F=ma relation is only suitable for objects with a constant mass!

Essentially, the formula defines a force to be a change in momentum, where momentum represents the mass of an object (m) of an object times its velocity (V). But how has this simple law been derived?

Imagine a car with mass (m) with a starting velocity of (V0) starting at point X0. The car speeds up and reaches another point (X1) at a velocity of (V1). Given that the mass of the car doesn't change and that force is equal to a change in momentum with a change of time, the formula becomes:

What about net force? Say you've slid a ball onto a smooth surface; your action of pushing is a force that isn't being cancelled out by other forces as the ball slides forward. The ball is experiencing acceleration. All the external forces have balanced out when the ball slows down and stops moving.

The gravitational force is the most common net force that makes an object move. If you throw a ball up in the air, the force of gravity pulls down on the ball once the ball loses upward momentum, and it will accelerate downwards. But gravity isn't the only force acting on an object; Newton's third law explains how to calculate net forces.

Newton's third law of motion states that whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first.

 For every action (force), there is an equal but opposite reaction.

 exerts a force on object Y, Y will also exert the same but opposite force on X. If this symmetry did not exist, a chair would collapse every time you sit on it. Your body exerts a force downward onto a chair, and the chair needs to exert an equal force upwards to remain standing.

 A normal force is a contact force that is perpendicular to the surface that an object is on. The normal force is always perpendicular to that object, whether you rest an object on a table or an angled ramp.

A normal force is special because it changes its magnitude. Say you have cling wrap over a deep dish and place one strawberry on it. The strawberry exerts a little force onto the cling wrap; a normal force from the cling wrap pushes right back with equal force. Add another strawberry, and the force due to gravity doubles, and so does the normal force. This will continue until you add enough strawberries that break through the cling wrap, i.e. when the normal force cannot match the force pushing against it.

The term 'for every action there is an equal but opposite reaction,' yet this kind of symmetry needs to explain why we can move things like picking up a cup of coffee. Well, things move because there's more going on other than just the action and reaction forces. There's motion as soon as forces become unbalanced (refer to Law #2).

, the app you're reading this one is a calculation management platform. You can sign-up and build hosted, shareable web apps (complete with an API and a web publishing module) with tools like Python and Spreadsheets. Learn more 

Explore Newton's laws of motion in our comprehensive guide. Enhance your understanding of the fundamentals. Discover more now!

Have you ever wondered why many structural elements like beams aren't commonly manufactured as fully-filled simple shapes like circles or squares?  Well, it's all about how efficiently a cross-section resists bending...and that's got a lot to do with a shapes Second Moment of Inertia (I).  

A column or a beam must typically contend with biaxial moments and flexural and shear forces. A member requires a minimum stiffness or, analogously, a minimum cross-sectional moment of inertia to resist these forces and avoid buckling or deflecting.  

So what exactly is the Second Moment of Inertia (I), what determines it, and how can engineers use this to their advantage? Let's take a look at an example to understand it fully.

Suppose you have a rectangular plank of wood and are using it to cross a void. You can orientate it flat, as shown on the right below, or rotate it vertically onto its side, as shown on the left.

Intuitively, you know that the arrangement on the left will perform better,  but the plank cross-sections and the area are the same, and so are its material properties. So what's changed? Well, by rotating the plank onto its side, its geometry and resultant moment of inertia have changed completely.

Some cross-sections are better at resisting bending than others, and the orientation of a shape has a major influence on a cross-section's ability to resist bending. This is precisely what the moment of inertia or the second moment of the area describes. When quantified, it measures how much resistance the cross-section has to bend. 

Let's unpack this example further by looking at the cross-sectional geometry in 2D.

The further the area of a material is concentrated away from its bending axis, also known as the neutral axis (N.A.), the stiffer the section is and the higher its moment of inertia will be, relative to that axis. Just by orientating an identical wooden plank on its side (left), the cross-section has more material located away from the (N.A.), thereby improving its resistance to bending. 

Take any small area in that plank of wood, it will either experience compressive or tension stresses, which increase linearly as you move up or down from the N.A. It will try to resist bending when subjected to a load by creating a moment around the neutral axis. This moment exerted increases linearly as you move away from the N.A. This means that the resistance to bending provided by any point in the cross-section is directly proportional to the distance from the neutral axis squared. Once you integrate the force times the distance from the N.A. on the entire surface, you get the total resistance to bending specific to the axis specified.

This is a critical phenomenon that we can use to our advantage. Let's take it further and add an I beam into the mix:

An I-beam maximizes the amount of material located far away from the NA. This gives it a bigger moment of inertia and greater stiffness for the same mass of material. As a result, it performs most efficiently in bending out of all three cross-sections shown. This is a reason why I-sections are so commonly seen in construction.

While the moment of Inertia (I) is a cross-sectional property that partially determines a section's resistance to bending and deflection, it also has many uses in other engineering applications.

The moment of inertia is used in beam and column analysis, calculating bending stresses and flexural rigidity. It is also important to note that it is not a fixed value that is unique to each cross-section; it changes depending on the location of the bending (or reference) axis. In general, most engineers will take the neutral axis x-x and y-y, located at the centroid of a cross-section profile, as the reference axis used to calculate the second moment of inertia.

How do you calculate the Second Moment of Inertia (I)?

Now that you understand what factors influence a cross-section's Second Moment of Inertia, we will outline the general steps required to calculate the moment of inertia.

1.  Understand the fundamental mathematical expressions

The Second Moment of Inertia is synonymous with the second moment of the area because it is calculated by taking the moment about the area twice: 

Now, you could explore the next steps in calculating the second moment of inertia by hand (which we recommend you do). Still, in case you find yourself pressed for time, we have compiled a summary of second moment of inertia formulas for common shapes and beam sections:

 Second moment of Area formulas for common shapes

2.  Break the cross-section up into segments

When calculating the Moment of Inertia, we must calculate the Moment of Inertia of smaller segments. Try to break them into simple rectangular sections or other sections for which the formula already exists. Here we can break this I-beam into three components:

Or this hollow rectangle into an equivalent outer rectangle minus the inner:

The Neutral Axis (N.A.) or the horizontal x-x axis is located at the centroid or center of mass. Most engineering cross-sectional shapes are symmetrical, so you will only need to find the x-x neutral axis.

Now let's combine the segments using the parallel axis theorem with the formula:

The second moment of inertia is one of the most important material properties of structural elements because it determines an engineer's section selections. Since this value is used to predict the resistance of beams to bending, torsion, and deflection, you'll see it pop up in quite a few engineering concepts and formulas, namely:

 the stress that an object encounters when it is subjected to a large load at a particular point, causing it to bend and become fatigued.

 Deflection is also known as displacement and can occur from externally applied loads or the structure's self-weight. If the deflection is large enough, you can visualize the degree to which an element/structure changes shape or is displaced from its original position- this is something we want to avoid at all costs!

Master the Fundamentals of Second Moment of Inertia. Enhance your engineering knowledge and optimize structural designs. Explore now

Second Moment of Inertia Fundamentals

The energy-momentum relation, also called the relativistic dispersion relation, is a foundational concept in physics, rooted in Albert Einstein's work on special relativity. 

This article will comprehensively overview the 

and present different equations for considering specific cases for this relation. 

It establishes an important connection between an object's energy and momentum, specifically relating its rest (intrinsic) mass, total energy and momentum.

This relation reveals the integral link between these physical properties. As an object's velocity approaches the speed of light, the interdependence of momentum and energy becomes evident, leading to consequences like time dilation and relativistic mass increase. 

The relation assumes the special relativity case of flat spacetime and that the particles are free. The total energy is the sum of the rest energy and 

. The invariant mass is the mass measured in the centre of the momentum frame, the inertial frame, whereby the total momentum of a system vanishes.

The energy-momentum equation is as follows:

Use this calculator to solve these equations!

Energy–momentum Relation Calculator

The energy-momentum relation is essential for understanding the behaviour of particles at high speeds, providing insights into the nuanced relationship between mass, energy, and motion in the framework of the universe.

If you liked this, check out our other articles and resources!

The energy-momentum relation, also called the relativistic dispersion relation, is a foundational concept in physics, learn about it here.