Mechanics Maths

What is kinematics?

Kinematics deals with displacement, time, velocity, and acceleration without considering the forces that cause the objects to move.

A simple example of this is the study of a car in motion. We can observe time, displacement, velocity, and acceleration.

A moving car will have a certain displacement . Recording two different moments when moving introduces the concept of time . When we combine the two, displacement over time, we have velocity . If the car isn't moving at a constant rate, the concept of acceleration (the change of velocity) comes in.

What is dynamics?

Dynamics is the area of mechanics that studies the forces that cause or modify the movement of an object. Dynamics is divided into linear dynamics and rotational dynamics. The first studies an object moving in linear motion, and the second studies objects that rotate around a fixed center, such as a chair in a carousel.

Dynamics works with concepts such as forces, the mass of the object in motion, its momentum (defined as the velocity multiplied by the object's mass), and energy.

Engineering requires you to apply the principles of mechanics from the point of view of kinematics or dynamics. Multiple applications range from the design of airplanes, bridges, cars, and buildings to the development of rockets for space exploration.

Quantities, units, and assumptions in mechanics

The study of mechanics is linked to quantities, which are the properties you can measure of an object. In an object in motion, the most important properties are the distance an object covers, the time it takes to cover this distance, the speed it has, how the speed changes, and the forces affecting the object.

The quantities to measure use Units. Units are standards used for each property we are measuring. Mechanics specifically uses the units for velocity (meters per second or m / s) and forces (Newtons), amongst others.

Another important aspect when dealing with mechanics is the simplification of the systems analyzed. These Assumptions allow you to study mechanics by reducing its complexity.

Physical quantities and units

In trying to understand what laws govern specific systems, we will need to quantify the physical elements that are going to be involved in the system.

Anything we can measure is known as a physical quantity . For example, if I say I weigh 80kg or the ruler is 30cm, you can assume 80kg is my mass, and 30cm is the length of the ruler. Every physical quantity must have two things:

• magnitude

• units

For example, if you say 20 kg of salt, 20 is the numerical value of the salt you have. This is not enough to conclude how much salt you have until the unit kg is added. The kilograms, or kg, is a SI unit - an international standard.

Units are necessary to specify the specific amount of what property of the substance we are measuring.

assumptions

Applying mathematics to real-life events can be complicated. There are so many variables it can be hard to know where to begin. You start by making the problem as simple as you possibly can.

There are certain things you can ignore, including:

• Air resistance.

• friction

• Energy dissipation.

• mass distribution.

It's helpful to know some keywords that are used for these Assumptions. For example, 'smooth surface' means there is no friction present on the surface, or if a particle has a ' negligible mass', it means you can assume its weight is zero.

Acceleration in kinematics

Remember, kinematics is an area of study that focuses on the movement of objects, disregarding the forces that cause the movements. This part of mechanics explores the concept of motion, and its relationship with time, velocity, and acceleration. The movements of the objects in kinematics can have a Constant Acceleration or a Variable Acceleration .

Constant acceleration and SUVAT equations

Constant Acceleration can also be called one-dimensional Equations for motion for Constant Acceleration. This employs the use of SUVAT Equations to find the values of any of the variables. SUVAT is an acronym of the variables to study. They are:

s, displacement in meters [m].

u, initial velocity in meters over seconds [m / s].

v, final velocity in meters over seconds [m/s].

a, acceleration in meters over seconds squared [m / s ² ] .

t, time in seconds [s].

Variable acceleration

In contrast to constant acceleration, Variable Acceleration primarily explores motion in objects where acceleration keeps changing. A variable acceleration means a variable velocity.

In mathematics, the formulations found to model the movement of an object are related to a mathematical area of study - Differentiation .

A typical example is to use the classical SUVAT formulation to calculate the acceleration from the displacement. The first derivation of the displacement will give you the velocity, and if you derive the velocity, you will obtain the acceleration.

If you are given the SUVAT formulation for the acceleration and want to find the displacement, you apply the inverse operation named Integration . Integrating the acceleration will give you the velocity, and if you integrate the velocity, you will obtain the displacement. Here are the equations:

Projectiles and parabolic motion

Projectiles and parabolic motion deal with objects projected through the air, describing a parabola during their movement. An example is throwing a ball.

This part of kinematics employs concepts of mathematics such as Trigonometry because of the Angles involved in the movements of the objects.

Figure 1.- Parabolic motion of a ball, showing the velocity components Vy and Vx

Forces and Newton's laws

Force can change the motion of an object. A straightforward way to describe force is as a pull or a push against an object. Newton's laws of motion and its mathematical expressions are central to how we describe forces every day.

These laws cover three significant ideas: the reciprocity of forces, the forces altering the state of movement of an object, and how mass, acceleration, and force relate to each other.

Another important aspect of the study of forces is how we use them to move objects and the mechanisms you can create to produce or affect them. Two examples of these mechanisms are Pulleys and moments produced by a bar.

Forces can also be present when an object has no movement; one example is the force of gravity on you as you remain standing. The study of forces when an object does not move ( in equilibrium ) or change its movement is called statics .

Newton's laws

Newton came up with three specific laws to describe the motion of an object.

• Newton's first law of motion states that an object continues to be in a state of rest or a state of motion at a constant speed along a straight line unless a force acting over the object changes this.

A ball will roll indefinitely if nothing stops it from moving. In this case, the friction against the air and the ground will cause it to stop.

• Newton's second law of motion states that the time rate of change of the momentum of a body is equal in both magnitude and direction to the force imposed on it. It can be modeled in an equation as:

$f=m·a$

Where f is the force in Newtons, m is the mass in kg, and a is the acceleration in m / s ² .

• Newton's third law of motion is also called the action and reaction forces law. It states that when a body exerts a force over another, the other body will exert a force equal in magnitude and opposite in direction.

An example is when you push against a hard wall, you will feel a push in the other direction.

pulleys

A pulley comprises a wheel and a fixed axle, with a groove along the edges to guide a rope or cable. It is not easy to lift heavy objects, so that is where Pulleys come in. Put two or more wheels together and run a wheel around them, and there you have an excellent lifting machine. The more pulleys you add to your machine, the more mechanical advantage you have at lifting a load easily.

statics

Statics deal with objects at rest and ones that are moving with constant velocity. In this object, forces are in equilibrium, so there is no change to its movement. One example of this is the forces over a building. The building structure is affected by gravity pulling it down, the force is distributed along the building, and the structure reacts to create an equilibrium.

friction

Friction is the force that resists the rolling and sliding of an object over a surface. Friction is a dissipative force, meaning that it can decrease the velocity of the objects in motion.

moments

A moment is a force you apply to something multiplied by the distance between the pivot and the force.

When a force is not enough to turn something around, you will need a pivot, too. Pivots and forces have a special relationship - if you push with the same force further away from the pivot, you can turn the item more easily due to a larger moment.

moment = force $\times$ distance

In a moment, the distance is the perpendicular distance to the point where you apply the force.