Planar Kinetics Of A Rigid Body: Force & Acceleration

Mass Moment Of Inertia:

The applied non-concurrent force system can cause the body to translate and rotate. The resistance of a body to angular acceleration is called Mass Moment of Inertia or simply Moment of Inertia ( M = Iɑ ).

• Mass moment of Inertia (MOI) is the integral of 2nd moment about any axis of all the element of mass 'dm'.
• For e.g: flywheel of engine.

• If the density of the body is not homogeneous, then you should use the following formula

Parallel Axis Theorem: 

If the moment of inertia passing through its own axis, then moment of inertia about any parallel axis is given by using Parallel Axis Theorem.

Moment of inertia can be reported about a specified axis as radius of gyration.

And, the moment of inertia for different shapes but most important of these are disk and rod.

If composite body is given and moment of inertia is required, then you calculate MOI as:

Planar Kinetic Equations Of Motion:

There are two types of kinetic equations which are described ahead:

1. Force Equation or Translational Equation Of Motion:

Force equation for the center of rigid body states that: The sum of all the forces acting on the body is equal to the body's mass times the acceleration of its mass center 'G'.

So, applying equation of motion:

2. Moment Equation or Rotational Equation Of Motion:

This equation involves the effect the rotation of rigid body. Applying 2nd condition of equilibrium on above object, we get the final result as:

Where, Mp represents kinetic moment. There are conditions for kinetic moments which are:

• If point P coincides with mass center G  → In this case 𝑥̅ = 𝑦̅ = 0. So above equation becomes ∑𝑀_𝐺 = 𝐼_𝐺𝛼.
• If there is no rotation, only translation  →  In this case, angular acceleration becomes zero ( α = 0 ),  ∑𝑀p = 𝑥̅m(ap)x - 𝑦̅m(ap)y.
• If there is both rotation and translation  →  In this case, the above equation remain as it was.

Equation Of Motion:

There are three cases for which equation of motion are to be defined:

1. Translation:

Straight line motion is called Translation. Curvilinear motion is considered as a sequence of rectilinear motion.

A. Rectilinear Translation:

There are two scenario along which analysis shall be completed and they are:

Considering mass center G as the starting point:

Considering point P as the starting point:

B. Curvilinear Translation:

There are two scenario along which analysis shall be completed and they are:

Considering mass center G as the starting point:

Considering point B as the starting point:

2. Rotation About A Fixed Axis:

There are two scenario along which analysis shall be completed and they are:

Considering mass center G as the starting point:

Considering point O as the starting point: (All equation remain the same however the Moment equation is changed as follows)

3. General Planar Motion:

This equation involves both translation and rotation. 

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.