Mathematics have many interesting fields, one of which is undeniably mechanics in maths. Mechanics is the study of the motion of objects and the forces behind that motion.
In essence, you will be dealing with lots of physics terms and theories, because that is where the knowledge of mechanics originate from. Newton’s laws, does it ring a bell? Well, it is a major part of this section of maths study, and any science student would tell you they first learned about Newton’s laws in their physics classes.
There is no surprise here, as both maths and physics are interrelated. You will notice that the moment you set foot in either one of the disciplines, calculating numbers is at their core.
Mechanics is an important topic that you must study in A-level maths, either the A level programs wherein you study for 9-10 months before writing an exam in order to seek admission directly into 200 level (such as IJMB), or the regular A level maths course in the university or other tertiary curricula in Nigeria.
In both cases, you are taught mechanics in mathematics. A topic, as important as calculus or logarithm in maths A level.
Some of the topics under mechanics include the following; kinematics, forces, Newton’s laws, friction, work energy and power, impulse and momentum, gravitation, vectors, center of mass, simple harmonic motion, energy and so much more.
To take on this particular field of study in maths, you need to have a solid background in algebra, and at least a passive knowledge of physics, because they will play a huge role when you want to solve mechanics in maths.
To begin, let us start by briefly revising some of the impressive concepts in this field of study. After that, we will walk you through how we solved the answer to an example question regarding forces in the article.
Gravity entails many things. What we all know is that it is the force keeping us to the ground, and that it brings whatever is thrown, back to Earth. For space shuttles to leave for space, it needs to break away from Earth's gravitational pull.
When something is free-falling, that is falling freely under the influence of gravity, it is subjected to gravitational acceleration, represented by a g. That is another phenomenon of gravity, often overlooked.
Acceleration is displacement per time spent. The displacement in gravitational acceleration is the falling distance, and the time is constant.
So to calculate the force of gravity an object is subjected to when falling, you say, g x mass of the object, since Force = mass x acceleration, F = ma, then the mass of the object multiplied by g is the gravitational force.
But instead of F for force, it transforms into Weight, W. So weight is actually a force. Next time you stand a scale trying to measure your weight, you should know that you are actually measuring the downward force you are exerting due to gravity.
The trigonometric ratio or trig ratio is a branch of maths that deals with geometric shapes, specifically right-angled triangles. Each right-angled triangle has one property that made it what it is, it is the fact that at least one of the angles is at a 90° in the triangle.
Trigonometry is used to solve for angles and length of sides, using a method that is abbreviated to SOHCAHTOA. Short for,
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent
You can read more about how the formulas came about and how you can use it to solve trig ratios from maths A level textbooks and from maths tutors.
In some more advanced maths questions, you will have to dissect a regular triangle into two to form a right-angled triangle. Solve for each part separately and connect the dots later.
We cannot really talk about mechanics and not talk about motion, it is the primary topic under this branch of mathematics in Nigeria. Motion is often described as the displacement of an object from point A to another point B.
The speed at which it is displaced and the direction matters a lot, so is the force directly or indirectly associated with the movement. The point is, when something is in motion, many factors are put into consideration.
Speed is a scalar quantity, in that it doesn’t have a direction, there is only mass or size, whereas vector and acceleration are vector quantities, they have both direction and size. All of these are quantities for measuring the speed at which an object is moving in time and space.
The path taken by a chased chicken is zigzagged and irregular, you should consider calculating its average speed when trying to find out how fast it is going, because it literally has no direction, it goes wherever it thinks it will be safe.
But when you want to find out how fast a car is going on a stretch of road that is straight, you can easily calculate the vector, it is similar to calculating for speed but this time the object (car in this case) has direction, a straight road.
Now, the rate of change of the vector of the car is known as acceleration, the rate of change is simply vector divided by the time taken, if you have covered differentiation and integration in your A level maths you will learn that it deals with calculations such as the one for acceleration in moving objects.
Motion is influenced by external forces, it is influenced by kinetic energy and by gravitational pull or energy and so much more, just like we have said it has countless factors that affect it and that must be put into consideration.
Example Of Mechanics In Math
Here is a sample question to shake things up a little bit. It is the kind of maths questions you are expected to solve when in mechanics. Do you think you can solve it without looking at the answers? Give it a try, once you are done, see if we got the same answers. Here’s the question;
If two forces are acting on a body, one acting horizontally at a magnitude of 50N and the other acting at an angle of 20° with a magnitude of 20N. What is the resultant force on the body?
Now, when solving maths questions like these always draw your diagram to get a visual representation of what you are doing and what it all meant. It is much easier to calculate the angles and other values if you can see them.
Since there are two forces traveling in different directions, we can’t simply add them up. We need to account for the direction of travel of these forces. The 20N force is acting at an angle of 20°, to begin we will need to separate it into its component forces.
That is the amount acting on the horizontal axis and the amount of force acting on the vertical axis. The 50N force is only in one direction so there is no need to break it down into its component forces, the amount acting on the vertical axis is zero.
From the diagram, we can complete it to form a right-angled triangle with the 20N force as our Hypotenuse at an angle of 20°. The opposite and adjacent sides of the triangle represent the number of forces on the vertical axis and horizontal axis respectively.
Using SOHCAHTOA, work out the adjacent side of the triangle which is our horizontal axis. Cosine will do.
Cosine = adjacent/hypotenuse
Cos 20 = adj/20
20 cos 20 = Adjacent.
Your answer is 18.8N, so the force acting in the horizontal direction of our 20N force is 18.8 Newton.
Moving on to the vertical component of the 20N force. We can use sine to find the opposite side of our triangle. So that sine of 20° will give us the value for Opposite divided by the hypotenuse.
Sine 20° = Opposite/Hypotenuse
Sine 20 = Opp/20
20 Sine 20 = Opp
The answer is 6.8N, meaning the opposite side is the side representing the vertical component of our 20N force that is acting on the body, and it is equal to 6.8N.
We know that we have a 50N force acting horizontally, and we calculated a force of 18.8N also acting horizontally, and a force of 6.8N acting vertically. The next step is to add them up.
Vertical forces first, you say,
6.8 + 0 = 6.8N
And for the horizontal forces, you say,
18.8 + 50 = 68.8N
We took the horizontal component of the 20N force and add it to the 50N force acting horizontally on the body. Finally, to find the overall resultant force, we create another triangle wherein the 68.8N represents the opposite side of the triangle i.e. the vertical component of the forces and the 6.8N represents the adjacent side of the triangle or the horizontal side.
Using the Pythagoras theorem which states that the hypotenuse is equal to √adjacent2 + opposite2, we can deduce the final answer. Substituting in our values we get,
Hypotenuse = √68.82 + √6.82 = 69.1N.
Your final answer is 69.1N, did you get it right?
Mechanics in maths is a fascinating topic. It reveals truths about the world around us that we were oblivious to, and shows us how it works.
Engineers, technicians, scientists and construction workers make use of mechanics to solve maths problems, and every one of us makes use of the principles of mechanics every day.