## What is Kinematics?

A machine is a device that accepts energy in some available form and utilizes it to do a type of work. Energy, work, or power has to be transferred from one mechanical part to another to run a machine. While the transfer of energy between two machine parts, those two parts experience a relative motion with each other. Studying such relative motions is termed kinematics.

## Why Kinematics is primarily studied?

If we want to design a car, what all parameters are there before making a real-time model? These parameters include:

• Design parameters
• Speed calculation
• Material selection
• Safety inspection
• Engine specifications
• Government law restrictions
• Pollution measurement
• Aesthetics
• Comfort
• Luxury
• Market analysis
• Cost estimation
• Time management
• Spare part assembly
• Design of various mechanical parts
• Variable speed reference.

There are multivariable parameters, before designing a car, all parameters cannot be considered in optimization, all cannot be handled by a single engineer, the work has to be portioned separately in running batch assembly and handed over to each of them. Design of joint and coupling for acceleration, velocity, deacceleration, angular velocity, torque, tension, etc. is also a part of mechanics, which helps in producing a relative motion for an object in less wear and tear. The study of kinematics in mechanics will help you to study the relative motion of a machine between various parts of the machine.

## Kinematics vs. Kinetics

In mechanics, the displacement of an object can be analyzed in various formulas. It may be vector addition, cross product, Newtonian analysis, assumption. This assumption plays a dominant role in the study of mechanics.

### Statics

It studies forces on the machine when it is at rest. Assuming the mass of such a machine is negligible.

### Dynamics

It deals with forces and their efforts, acted upon by any machine parts in motion. All parts causing motion will not be considered to avoid complexity, some parameters are assumed to be negligible. We come to kinetics and kinematics

### Kinematics

It deals with geometries in motion and concepts like displacement, velocity, and acceleration. From where the force has originated is not an issue in kinematics, when it reaches our geometry, what relative force it creates or has to be created is considered.

### Kinetics

The cause and effect of the motion of an object, even if it is starting point- inertia is considered. The force that comes into action and the force we convert into useful motion both are considered for calculation in kinetics.

## The Motion of an object in Kinematics

When a body is fixed with a constrained motion in a single plane is known as plane motion. The plane motion can be either rectilinear or curvilinear depending on the part application.

Rectilinear motion : If a motion of an object can travel in a straight-line path, it is termed rectilinear or translatory motion. E.g. – piston, hydraulic lift

Curvilinear motion : In a single plane, if motion is confined to a curved path, it is curvilinear motion. E.g. – flywheel, slider-crank, gear trains.

## Displacement

Whenever an object is in variable motion in kinematics, it gets displaced from one position to another. Displacement of a body is considered to be a vector quantity; hence it possesses both direction and magnitude. This physical quantity can either be linear or angular.

Imagine a reciprocating petrol engine in an automobile; the parts such as piston, piston rod, and the cylindrical head will displace in a linear path, whereas flywheel, crankshaft, crankpin, wheels, disc brakes will displace in circular or angular motion.

The displacement calculation in kinematics can be done by an engineer in two methods, although we have technology incorporated software tools nowadays, it also runs and compares only on this basic method.

• Equation or derivation method (calculative on physical laws)
• Graphical method (tracing of parameter interrelation)

We will discuss velocity, acceleration, translational velocity, translational acceleration, angular velocity, and angular acceleration in the displacement of kinematics. We will analyze in a derivational method.

## Linear Velocity

A rate of change of displacement distance of a body with respect to time is termed linear velocity. With linear velocity we can predict the direction and distance, hence it is a vector quantity. It is denoted as v.

$\begin{array}{c}v=\frac{displacement-dis\mathrm{tan}ce}{time}\\ =\frac{ds}{dt}\\ =\frac{m}{s}\end{array}$

## Linear Acceleration

A change in a velocity of a body with respect to time is known as linear acceleration. A difference in final velocity and initial velocity with respect to time is known as linear acceleration. It is also vector quantity, it is denoted by a.

a= (final velocity – initial velocity)/ time

$\begin{array}{c}a=\frac{v-u}{t}\\ =\frac{dv}{dt}\\ =\frac{d}{dt}\left(\frac{ds}{dt}\right)\\ =\frac{{d}^{2}s}{d{t}^{2}}\end{array}$

Its dimension is m/s2

## Equations of linear motion

There is some interrelation between displacement velocity, acceleration, and time. The following will be the equation of linear motion. The useful derivation will be found in engineering mechanics.

Formula

$\begin{array}{c}v=u+at\\ s=ut+1}{2}a{t}^{2}\\ {v}^{2}={u}^{2}+2as\end{array}$

u - initial velocity of the body

v - final velocity of  the body

a – acceleration of the body

s- displacement distance of a body

t – time

## Angular Displacement

If a body is displaced in restricted rotational motion or angular motion, it is termed angular displacement. It has both magnitude and direction; hence it is a vector quantity.

## Angular Velocity

A rate of change of angular displacement of an object is termed angular velocity. It is denoted by letter omega (ω)

$\omega =d\theta /dt$

It is also a vector quantity denoted by rad/s

Angular acceleration - definition

A definition of angular acceleration denotes that it is the rate of change of angular velocity with respect to time. It is denoted by α(alpha).

$\begin{array}{c}\alpha =\frac{d\omega }{dt}\\ =\frac{d}{dt}\left(\frac{d\theta }{dt}\right)\\ =\frac{{d}^{2}\theta }{d{t}^{2}}\end{array}$

## Derivational Equations of Angular Motion

Angular velocity, angular acceleration, and angular displacement are interrelated by some physical law in mechanics.

Formulas:

$\begin{array}{c}\omega ={\omega }_{0}+\alpha t\\ {\omega }^{2}={\omega }_{0}{}^{2}+2\alpha \theta \\ \theta ={\omega }_{0}t+1}{2}\alpha {t}^{2}\\ \theta =\left({\omega }_{0}+\omega \right)t/2\end{array}$

The most important part of kinematics is the relationship between linear motion and angular motion. Imagine the petrol engine we stated at the start of the subject, reference of that engine, it will convert linear motion into angular motion. The linear acceleration of the piston is converted into an angular acceleration in the crankshaft. There is a relationship between various motions, it's an important definition to define the relation between linear and angular quantities of dynamics.

## Relation between Linear and Angular Quantities of Motion in Kinematics

Consider an object moving in a circular path from A to B as shown in the figure

r= radius of the circular path

θ = angular displacement in radius

S = linear displacement

V= linear velocity

ω - angular velocity

a- linear acceleration

α - angular acceleration

By geometry

S = r θ

Linear velocity

$\begin{array}{c}v=\frac{ds}{dt}\\ =\frac{d\left(r,\theta \right)}{dt}\\ =r×\frac{d\theta }{dt}\\ =r\omega \end{array}$

Linear acceleration

$\begin{array}{c}a=\frac{dv}{dt}\\ =\frac{d\left(r,\omega \right)}{dt}\\ =r×\frac{d\omega }{dt}\\ =r×\alpha \end{array}$

The relative motion of kinematics between acceleration, and angular acceleration, velocity and angular velocity is a basic step of kinematics standing as a tool to all major mechanical parts working in motion.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate

courses, especially for

• Bachelors in Science (Physics)
• Masters in Science (Physics)

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