August 22, 2021

Applying Integrals to a Solid Body

What is work? In physics, work is defined simply as the effort exerted on an object by some other object, which causes the object to alter its internal state. The sum of all such efforts is the work done. If the distance between the two objects is fixed, then the total work done is simply the product of the constant force acting on the object and the time it takes to go from point A to point B.


There are many ways in which this concept can be applied, in specific cases being transportation. Let’s say that you are driving down the highway and you hit a pothole or a bump in the road. Your car stops, and you have to swerve suddenly to avoid hitting it, and you do so, but since your car did not stop at all, there is now work done to move the vehicle into the right position to prevent it from crashing. Now if this accident happens again, then the first accident would have been worth less than the second, because the second accident would have been considered a zero-sum game, where in the first accident, more work had to be done to cause the outcome, and so the total amount of work done would have been smaller.

This brings us to the question of how the law of conservation of energy applies to our world. The work done to move the particles from one state to another is referred to as the work done, and if they are not replaced, then that energy is eventually going to be lost. Now the only thing that prevents the particles from being replaced is the principle of relativity called the law of conservation of energy. Conservation of energy tells us that the total energy, no matter what that energy is, will always remain the same, thus there is no way for the energy to be added or subtracted. Thus the only solution is to move along with the constant speed of the particles, and this is done by your car’s wheels.

The first thing we want to know is why the velocity of anything is referred to as a velocity. Well this is simple enough to understand; the way in which you describe motion is through the transfer of a constant force. Now the law of integrals shows us that if the system has a constant force that acts on a single variable, then the resultant of those actions will be a single variable. The example that I am using here is that if you take your hand and try to lift a piano lid by applying a constant force to it, this will always result in two separate variables, but if you act to lift the piano lid by doing the same action as before, then the resultant of those actions will become a single variable, which we will label “piano lid velocity”.

There are many ways in which we can calculate this integral, and one of them is the integral of a constant variable. To find the integral, we must take a look at the work function, where we differentiate between the time and the position for each individual work – in our case, we are taking a look at the instantaneous torque we are applying to the piston. Once we have differentiated this, we must find the derivative of time, which is the integral of the constant t; this is done by looking at the tangent of the velocity vector. This should give us our integral, so we now just need to apply it to our work function and solve for t. If we do this carefully, we should get another derivative, namely the torque that we are acting on the piston, and this gives us the final term of our integral; torque(velocity vector). This tells us that the angle of attack of our vehicle determines the torque that we apply to it, and the larger the angle, the greater the torque.

There are many other ways in which we can use the integral of a constant and the trajectory of a rigid body, such as for the case of springs. You may also find applications for the other components of a rigid body such as the dynamic or kinetic energy of an object or the momentum that it has after being deformed, among many others. For these purposes, I would advise you to compute the integral of your function on the computer and then plot it onto a graph, as this will give you a nice visual representation of what you are looking at. The integration of rigid bodies is only possible if you have a good understanding of the language of dynamics and mathematics, and I hope that this introduction has shown you just that. Good luck!