Kinetics Of A Rigid Body: Work & Energy, Impulse & Momentum

Work And Energy:

If a force acts on a body and displaces it, then the body has some Work done. The capability of  a body to do work is called Energy. Some important principles and topics related to work and energy are described ahead.

Principle Of Work & Energy:

Work and energy theorem as in particle kinetics is given by

The principle of work and energy depends on whether the body is translating, rotating and general planar motion. There are three conditions which are:

• If body is Translating  →  Kinetic energy equals to Translational Kinetic energy ( T = ½ mv² ).
• If body is Rotating  →  Kinetic energy equals to Rotational Kinetic energy ( T = ½ Iω² ).
• If body is in General Plane Motion  →  Kinetic energy is equal to the sum of Translational and Rotational kinetic energies ( T = ½ mv² + ½ Iω² ).

Work Of A Variable Force:

If the force for work done is variable depending upon its path, The formula for work done is given by:

Work Of A Constant Force:

If the force for work done is constant along the path, the formula for work done is given by:

Work Of A Weight:

The work due to weight is done only when the body undergo vertical displacement under action of gravity. The formula for the work done by weight is given by:

Work Of A Spring Force:

The work done by spring force is given by:

Work Of A Couple Moment:

Couple is defined as two parallel force having the same magnitude and opposite direction and is separated by perpendicular distance. For planar kinetics, moment vector and angular position vector are both perpendicular to plane, so they can have same or opposite directions.

Conservation Of Energy:

If the work due to non-conservative forces is zero, then the equation of work and energy is given by:

Impulse And Momentum:

The momentum for shorter period of time or very large contacting force acting for very short period of time between bodies or surfaces is called Impulse.

Linear Momentum:

The quantity of motion of a body or the product of mass and velocity of a body is called Linear Momentum.

Angular Momentum:

The quantity of rotation of a body or the product of moment of inertia and angular velocity is called Angular Momentum.

There are three cases of Angular and Linear Momentum which are described as follows:

1. Translation:

If the body is subjected to rectilinear and curvilinear translation, then angular velocity (⍵) is zero. So the linear and angular momentum of body is given by:

2. Rotation About A Fixed Axis:

If the body is subjected to rotation, then linear velocity (v) is zero. So the linear and angular momentum of body is given by:

3. General Planar Motion:

If the body is subjected to rotation as well as rectilinear or curvilinear translation then, the linear and angular momentum of body is given by:

Principle Of Linear Impulse & Momentum:

Principle of linear impulse and momentum states that: The initial momentum of body gained some additional momentum due to collision or contact with other surface which we called impulse for final momentum. Mathematically;

Principle Of Angular Impulse & Momentum:

Principle of angular impulse and momentum states that: The initial angular momentum of body gained some additional angular momentum due to collision or contact with other surface which we called angular impulse for final angular momentum. Mathematically;

Conservation Of Momentum:

• If there is no (negligible) net external linear impulse, then;

• If there is no (negligible) net external angular impulse, then;

• If you want to know about gyroscope, its motion and couple and something about Eccentric Impact, then click here.
• Dynamics behavior of engineering system can be described as:

• Static and Dynamic Analysis of Skyscrapers:

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.