If the moment of inertia of a body along a perpendicular axis... | Agrigyan

Detailed Derivation of Centre of Gravity of a Solid Cone:

• Integration along axis:
  Consider a solid right circular cone of base radius $r$ and total height $h$. Take an elemental disk of thickness $dy$ at distance $y$ from the apex:
  $$\text{Radius of disk } x = \frac{r}{h} y, \quad dV = \pi x^2 dy = \pi \left(\frac{r}{h}\right)^2 y^2 dy$$
  $$\bar{y}_{\text{apex}} = \frac{\int_0^h y dV}{\int_0^h dV} = \frac{\pi \frac{r^2}{h^2} \int_0^h y^3 dy}{\frac{1}{3} \pi r^2 h} = \frac{\frac{h^4}{4}}{\frac{h^3}{3}} = \frac{3}{4} h$$
• Distance from the Base:
  $$\bar{y}_{\text{base}} = h - \bar{y}_{\text{apex}} = h - \frac{3}{4} h = \frac{h}{4}$$

Detailed Step-by-Step Derivation:

• Centroidal Moment of Inertia ($I_G$):
  For a triangle of base $b$ and height $h$, about an axis through its centre of gravity parallel to base:
  $$I_G = \frac{b h^3}{36}$$
• Moment of Inertia About Vertex Axis ($I_{\text{vertex}}$):
  The distance from the centroid to the vertex is $d = \frac{2}{3} h$. Using the parallel axis theorem:
  $$I_{\text{vertex}} = I_G + A d^2 = \frac{b h^3}{36} + \left(\frac{1}{2} b h\right) \left(\frac{2}{3} h\right)^2$$
  $$I_{\text{vertex}} = \frac{b h^3}{36} + \frac{b h}{2} \left(\frac{4 h^2}{9}\right) = \frac{b h^3}{36} + \frac{2 b h^3}{9} = \frac{b h^3 + 8 b h^3}{36} = \frac{9 b h^3}{36} = \frac{b h^3}{4}$$
• Ratio:
  $$\frac{I_{\text{vertex}}}{I_G} = \frac{\frac{b h^3}{4}}{\frac{b h^3}{36}} = \frac{36}{4} = 9$$

Detailed Analysis of Physical Units:

• Work: Defined as Force $\times$ Displacement, unit is $\text{N}\cdot\text{m}$ (or Joule). (Included).
• Energy: Capacity to do work, dimensionally identical to work, unit is $\text{N}\cdot\text{m}$ (or Joule). (Included).
• Torque: Defined as Force $\times$ Perpendicular distance, unit is strictly $\text{N}\cdot\text{m}$. (Included).
• Power: Rate of doing work, unit is Watt ($\text{J/s} = \text{N}\cdot\text{m/s}$). (Not N-m).
• Momentum: Mass $\times$ Velocity, unit is $\text{kg}\cdot\text{m/s}$ or $\text{N}\cdot\text{s}$. (Not N-m).