Rotational Motion, Mass Moment of Inertia and Radius of Gyration

1. Rotational Motion can be of 2 types
   1. Spinning Motion of an Object along any Axis that Passes through its Center of Mass.
   2. Orbiting Motion of an Object along any Axis that Does Not Pass through its Center of Mass.
2. A Principal Axis of Rotation for any Object is an Axis passing through its Center of Mass along which if the Object Rotates, the Direction of its Spin Angular Momentum is Parallel to that of its Spin Angular Velocity. Every Solid Rigid Body Object has atleast One Set of 3 Mutually Perpendicular Principal Axes known as the Principal Axes of Rotation or Principal Axes of Inertia. For ease of calculation , these 3 Axes are considered to be aligned to the \(X\), \(Y\) and \(Z\) Axes of the Cartesian Coordinate System.
3. For Equi-Symmetric Objects (i.e. Objects that have their Mass and Volume Symmetrically Distributed Accross 3 Mutually Perpendicular Axis such as Sphere, Cube etc.), Every Axis of Rotation Passing through its Center of Mass is a Principal Axis. For these Objects Any Set of 3 Mutually Perpendicular Axes can be Principal Axes of Rotation or Principal Axes of Inertia.
4. Just like Mass of an Object provides Resistance to Change in its Linear Motion, Mass Moment of Inertia of an Object provides Resistance to Change in its Rotational Motion. From hereon in this document Mass Moment of Inertia shall be just referred as Moment of Inertia.
5. Corresponding to 2 types of Rotational Motion, Moment of Inertia for any Object is also of 2 types
   1. Spin Moment of Inertia of Object along any Axis that Passes through its Center of Mass.
   2. Orbital Moment of Inertia of Object along any Axis that Does Not Pass through its Center of Mass.
   The Total Moment of Inertia of any Object is Sum of its Spin and Orbital Moments of Inertia.
6. The Total Moment of Inertia of any Object depends on
   1. Mass of the Object
   2. Spin Moment of Inertia Depends on Orientation of the Axis of Rotation With Repect to the Object.
   3. Orbital Moment of Inertia Depends on Distance of the Axis of Rotation from the Center of Mass of the Object.
7. The Moment of Inertia of any Object is either represented as a Scalar Value (Denoted By \(I\)) or a \(3\times 3\) Matrix called Moment of Inertia Tensor (Denoted By \(\overleftrightarrow{I}\)). The Different Types of Moment of Inertia are Denoted as follows
   1. Spin Moment of Inertia along Principal Axes of Inertia is Denoted as \(I_p\) (Scalar Value) or \(\overleftrightarrow{I_p}\) (Tensor)
   2. Spin Moment of Inertia along Any Axis Other than Principal Axes of Inertia is Denoted as \(I_s\) (Scalar Value) or \(\overleftrightarrow{I_s}\) (Tensor).
   3. Orbital Moment of Inertia is Denoted as \(I_o\) (Scalar Value) or \(\overleftrightarrow{I_o}\) (Tensor).
   4. Total Moment of Inertia is Denoted as \(I_t\) (Scalar Value) or \(\overleftrightarrow{I_t}\) (Tensor). Also as mentioned above
      
      \(\overleftrightarrow{I_t}=\overleftrightarrow{I_s} + \overleftrightarrow{I_o}\)   and   \(I_t=I_s + I_o\)
   The Moment of Inertia must be represented as Tensor for accurate calculation of Direction and Magnitude of Spin Angular Momentum and Corresponding Torque (except for Equi Symmetric Objects for which Scalar Value can be used Interchangably).
   
   The Scalar Value of any type of Moment of Inertia can be calculated from its Corresponding Tensor as follows
   
   \(I=\hat{n}^T\overleftrightarrow{I}\hat{n}\)
   
   where \(\hat{n}\) is the Unit Vector along Direction of the Axis of Rotation (i.e. Angular Velocity).
8. The Spin Moment of Inertia Tensor of any Object along its Principal Axes of Inertia \(\overleftrightarrow{I_p}\) is a \(3\times 3\) Diagonal Matrix given as follows
   
   \(\overleftrightarrow{I_p}=\begin{bmatrix}I_{x} & 0 & 0 \\ 0 & I_{y} & 0 \\ 0 & 0 & I_{z} \end{bmatrix}\)
   
   Each Scalar Value along the Diagonal of this Matrix (\(I_{x}\), \(I_{y}\) and \(I_{z}\)) gives the Value of Moment of Inertia along One of the Principal Axes and each value is called a Principal Moment of Inertia.
   
   \(I_x\) is the Principal Moment of Inertia along the Principal Axis of Object that is Aligned with the \(X\) Axis of Cartesian Coordinate System
   
   \(I_y\) is the Principal Moment of Inertia along the Principal Axis of Object that is Aligned with the \(Y\) Axis of Cartesian Coordinate System
   
   \(I_z\) is the Principal Moment of Inertia along the Principal Axis of Object that is Aligned with the \(Z\) Axis of Cartesian Coordinate System
   
   For Equi-Symmetric Objects, all the Principal Moments of Inertia are Equal (i.e. \(I_{x}=I_{y}=I_{z}=I\)). Hence, the Spin Moment of Inertia Tensor for such Objects is a \(3\times 3\) Scalar Matrix and for all calculations, the Spin Moment Of Inertia for all Equi-Symmetric Objects can be taken as a Scalar Value.
9. The Spin Moment of Inertia Tensor of any Object along Any Axis Other than Principal Axes of Inertia (\(\overleftrightarrow{I_s}\)) is calculted as follows
   
   \(\overleftrightarrow{I_s}=R\overleftrightarrow{I_p}R^T\)
   
   where \(R\) is the Rotation Matrix Corresponding to the Axis of Rotation. The following gives the relation between \(\overleftrightarrow{I_s}\) and \(\overleftrightarrow{I_p}\)
   1. The Spin Moment of Inertia Tensor \(\overleftrightarrow{I_s}\) is a Symmetric Matrix Similar to the Spin Moment of Inertia Tensor \(\overleftrightarrow{I_p}\) given as follows
      
      \(\overleftrightarrow{I_s}=\begin{bmatrix}I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{bmatrix}\)    where   \(I_{xy}=I_{yx}\),   \(I_{xz}=I_{zx}\)   and   \(I_{yz}=I_{zy}\)
   2. \(\overleftrightarrow{I_p}\) is the Eigen Value Diagonal Matrix for all possible Spin Moment of Inertia Tensors \(\overleftrightarrow{I_s}\) of an Object.
   3. For Equi-Symmetric Objects \(\overleftrightarrow{I_s}=\overleftrightarrow{I_p}\) for All Angles of Rotation for Any Axis. For other Objects \(\overleftrightarrow{I_s}=\overleftrightarrow{I_p}\) only when Rotation Angle is 0.
   4. For any Object \(I_s=I_p\) (i.e the Scalar Values of Spin Moment of Inertia are same) for all Angles of Rotation for any Axis.
10. The Scalar Value of Orbital Moment of Inertia (\(I_o\)) of any Object of Mass \(M\) is calculated as
    
    \(I_o=MD^2\)
    
    where \(D\) is the Perpendicular Distance between the Center of Mass of the Object and the Axis of Rotation. The quantity \(D^2\) is actually the Square of the Length of Displacement Vector \(\vec{D}\) between Center of Mass of the Object and the Axis of Rotation. Given Displacement Vector \(\vec{D}\), the following shows how the Orbital Moment of Inertia Tensor \(\overleftrightarrow{I_o}\) is calculated
    
    \(\vec{D}=\begin{bmatrix}x\\y\\z\end{bmatrix}\hspace{1cm}\overleftrightarrow{D^2}=\begin{bmatrix}y^2+z^2 & -xy & -xz \\ -yx & x^2+z^2 & -yz \\-zx & -zy & x^2+y^2\end{bmatrix}\)
    
    \(\overleftrightarrow{I_o}=M\overleftrightarrow{D^2}=M\begin{bmatrix}y^2+z^2 & -xy & -xz \\ -yx & x^2+z^2 & -yz \\-zx & -zy & x^2+y^2\end{bmatrix}=\begin{bmatrix}My^2+Mz^2 & -Mxy & -Mxz \\ -Myx & Mx^2+Mz^2 & -Myz \\-Mzx & -Mzy & Mx^2+My^2\end{bmatrix}\)
11. Dimensionless Objects such as Point Masses only have Orbital Moments of Inertia and Do Not Have any Spin Moments of Inertia.
12. Any Object can be considered to be made up of a Large Number of Point Masses (Possibly Infinite) distributed accross space. Hence the Total Moment of Inertia of any Object of Mass \(M\) along any Axis is actually a Sum of Orbital Moments of Inertia of all those Point Masses that make up the Object around that Axis as given in the following
    
    \(I_A=m_1{D_1}^2 + m_2{D_2}^2 + ... + m_n{D_n}^2\)
    
    where
    
    \(I_A=\)Total Moment of Inertia of the Object around an Axis \(A\)
    
    \(m_1, m_2, ..., m_n=\) Point Masses that make up the Object
    
    \(D_1, D_2, ..., D_n=\) Perpendicular Distance of Corresponding Masses from the Axis \(A\)
    
    Now, If if all the Point Masses that make up the Object are equal i.e. \(m_1=m_2=...=m_n=m\), we have
    
    \(I_A=m({D_1}^2 + {D_2}^2 + ... + {D_n}^2)\)
    
    Also, Since
    
    \(M=m_1 + m_2 + ... + m_n\hspace{.5cm}\therefore M=n \times m \hspace{.5cm}\Rightarrow m= \frac{M}{n}\)
    
    Therefore,    \(I_A=M \frac{({D_1}^2 + {D_2}^2 + ... + {D_n}^2)}{n}\)    \(\Rightarrow \frac{I_A}{M}=\frac{({D_1}^2 + {D_2}^2 + ... + {D_n}^2)}{n}\)    \(\Rightarrow \sqrt{\frac{I_A}{M}}=\sqrt{\frac{({D_1}^2 + {D_2}^2 + ... + {D_n}^2)}{n}}=R_g\)
    
    The value of the expression \(\sqrt{\frac{I_A}{M}}\) is called the Radius of Gyration (\(R_g\)) of the Object around the Axis \(A\).
    
    Radius of Gyration for any Object around any Axis of Rotation is defined as Square Root of the Average of Square of Distances (or Root Mean Square Distance) of All the Point Masses that make up the Object from that Axis.
    
    The following are 2 important observations based on above calculations
    1. The Total Moment of Inertia of any Object around any Axis is the Product of its Mass and Square of its Radius of Gyration around that Axis.
    2. Except when specified for the a Single Point Mass, Radius of Gyration is a Scalar Quantity (since its an Average).
13. The values of Principal Moments of Inertia (\(I_x\), \(I_y\) and \(I_z\)) for any Object are calculated by Integrating the Orbital Moments Inertia of Point Masses that make up the Object around the corresponding Principal Axes of Inertia.