Kinematics Of A Particle

Mechanics:

It is the branch of physics which deals with forces acting on a body at rest or motion.

Dynamics:

It is the branch of mechanics in which we study motion of bodies and forces which are doing this.

Dynamics are further divided into two:

• Kinematics  →  It tells about geometric aspect of motion.
• Kinetics  →  It deals with the analysis of forces causing motion.

Following definitions are important to describe the motion of a body:

• Displacement  →  The shortest possible path followed by a particle.
• Speed  →  Distance per unit time is called Speed.
• Velocity  →  Displacement per unit time is called Velocity.
• Average Velocity  →  Algebraic sum of displacements per unit change in time.
• Average speed  →  Mathematical sum of distances per unit change in time.
• Instantaneous Velocity  →  Velocity at any instant when time tends to zero or velocity for very short interval of time.
• Acceleration  →  Change in velocity with respect to change in time.
• Instantaneous Acceleration  →  Acceleration at any instant when time tends to zero or acceleration for very short interval of time.

Irregular Motion or Erratic Motion:

The position time dependence cannot be fully captured by only one time period but use time intervals.

There are three types of kinematic graphs which are as follows:

1. For Change in Velocity, you should have a-t graph.

2. For displacement, you should have v-t graph.

3. For acceleration, you have constant velocity times the slope of v-s graph.

To calculate velocity, acceleration and displacement watch the video given below:

General Curvilinear Motion:

The motion of body along a curved path is called general curvilinear motion.

• If change in time is very small, line between points becomes straight and form tangent to position vector.
• Osculating Plane  →  refers to the plane which contains arc on which we are focusing on.
• The 3D motion is considered as a sequence of 2D motion. 

General curvilinear motion is described by three coordinate systems which are given below:

1. By Rectangular Components:

These rectangular components dependence on the instant position of particle locus. Only use this coordinates system when the path followed by a particle is known or its coordinates are known.

The velocity and acceleration for curvilinear motion using rectangular coordinates system is given by:

2. By Normal & Tangential Components:

Always use these coordinates when path of particle is known and there is homogeneity in the curve. The n-axis always act towards the center of curvature 'O'.

The acceleration of the particle following a curve path is given by:

Where, tangential and normal components of acceleration are:

There are two cases which is produced by changing the curve or making constant velocity of particle which are:

1. Particle moves along a Straight line  →  radius of curvature becomes infinity (⍴ → ∞) which makes a_n = 0.
2. Particle moves along a curve with constant speed  →  change in velocity with respect to time becomes zero (a_t = 0).

3. By Cylindrical Coordinates:

The motion of particle is also described by cylindrical coordinates in which:

• Radial coordinate which have direction outward from the origin.
• Transverse coordinate which have counter-clockwise angle between reference line and radial axis.

The velocity of particle following the curve path is given by:

And, the acceleration of particle is given by:

Motion Of Projectile:

It is the motion of an object by its own inertia by the downward force of gravity. Main points for projectile motion is given by:

• Motion is along curved path and under action of gravity.
• Equations of motion are applied.

Dependent Motion:

The motion of bodies when one body moves relative to other or when both bodies are interconnected is called Dependent Motion.

• It uses single fixed reference frame.

There are two cases to describe dependent motion which are as follows:

1. Analysis Of Two Interconnected Particles:

To calculate the motion of interconnected particles, follow these steps given below:

• Generate Datum.
• Represent the position.
• Recognize the constant lengths.
• Describe depending relations between position variables.  

2. Relative Motion Using Translating Reference Frame:

Translating frame of reference is describe by x', y', z' and fixed frame by x, y, z. The position vectors are related to each other by:

For velocity differentiate with respect to time and for acceleration differentiate twice, we get:

• Why do we learn Dynamics? Watch to find the question's answer:

• In this course we only learn Linear dynamics, if you are interested into dynamics have a little overview of Non-Linear Dynamics:

References:

• Material from Class Lectures + Book named Engineering Mechanics Statics & Dynamics by R.C. Hibbeler (12th Edition) + from YouTube channel named Yiheng Wang for Engineering Dynamics Mechanics + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.