Shortcuts and Tricks:

1. Parallel Axis Theorem: For an object of mass M, distance D from an axis of rotation, and moment of inertia I about the COM:

• The moment of inertia about the parallel axis is: $$I_{parallel} = I_{COM} + MD^2$$

2. Perpendicular Axis Theorem: For an object with COM moments of inertia (I_x,I_y,I_z) and moment of inertia (I_{com}) about its COM:

• For rotation about axes perpendicular to one of the principal axes, the moment of inertia is given by: $$I_{perpendicular} = I_{COM} + I_x + I_y + I_z$$

3. Rods:

• The moment of inertia of a uniform rod of length (L) and mass (M) about its COM is: $$I_{COM}=\frac{1}{12}ML^2$$
• The moment of inertia about an end is given by:$$I_{end}=\frac{1}{3}ML^2$$

4. Rectangular Plates:

• The moment of inertia of a rectangular plate of length (L), width (W), and mass (M) about its COM is: $$I_{COM}=\frac{1}{12}M(L^2+W^2)$$

5. Hollow/Solid Cylinders:

• For a hollow cylinder with radius (R), height (H), outer radius (R_0), inner radius (R_i) and mass (M): $$I_{COM}=\frac{1}{2}M(R_i^2+R_0^2)$$
• For a solid cylinder with radius (R) and mass (M): $$I_{COM}=\frac{1}{2}MR^2$$

6. Spheres:

• For a sphere with radius (R) and mass (M): $$I_{COM}=\frac{2}{5}MR^2$$

CBSE Board Exam Numericals:

1. Rod: (M = 2\texttt{ kg}, L = 1\texttt{ m}, I_{COM}=\frac{1}{3}\texttt{ kg m}^2). Solution: $$I_{end}=I_{COM}+Md^2=\frac{1}{3}\texttt{ kg m}^2+(2\texttt{ kg})(1\texttt{ m})^2=\frac{7}{3}\texttt{ kg m}^2$$

2. Square Plate: (M = 4\texttt{ kg}, L = 2\texttt{ m}, I_{COM}=\frac{8}{3}\texttt{ kg m}^2). Solution: $$I_{parallel}=I_{COM}+Md^2=\frac{8}{3}\texttt{ kg m}^2+(4\texttt{ kg})(2\texttt{ m})^2=\frac{32}{3}\texttt{ kg m}^2$$

3. Solid Sphere: (M = 6\texttt{ kg}, R = 1\texttt{ m}, I_{COM}=\frac{2}{5}\texttt{ kg m}^2). Solution: $$I_{tangent}=I_{COM}+MR^2=\frac{2}{5}\texttt{ kg m}^2+(6\texttt{ kg})(1\texttt{ m})^2=\frac{12}{5}\texttt{ kg m}^2$$

JEE Exam Numericals:

1. Uniform Rod: (M,L), (I_{COM}=\frac{1}{12}ML^2).

• I about the axis perpendicular to the rod and passing through its end: $$I_{end}=\frac{1}{3}ML^2$$

2. Square Plate: (M,L), (I_{COM}=\frac{1}{6}ML^2).

• I about any other axis perpendicular to the plate: $$I_{\perp}=\frac{1}{3}ML^2$$

3. Solid Sphere: (M,R), (I_{COM}=\frac{2}{5}MR^2)

• I about any other axis: $$I_{other}=\frac{7}{5}MR^2$$

4. Thin Rod: (M,L), (I_{COM}=\frac{1}{12}ML^2).

• I about an axis perpendicular to the rod and passing through one end: $$I_{end}=\frac{1}{3}ML^2$$

5. Thin Hoop: (M,R), (I_{COM}=MR^2)

• I about any other axis perpendicular to the hoop passing through its center: $$I_{other}=2MR^2$$