Mass Moment and Centre of Mass:

• Mass Moment: $$\vec{M} = m \vec{r}$$

• Centre of Mass of a System of ‘N’ Discrete Particles: $$\vec{r}_{cm} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + \ldots + m_n\vec{r}_n}{m_1 + m_2 + \ldots + m_n}$$

• Centre of Mass of a Continuous Mass Distribution: $$x_{cm} = \frac{\int x dm}{\int dm}, \quad y_{cm} = \frac{\int y dm}{\int dm}, \quad z_{cm} = \frac{\int z dm}{\int dm}$$

$$\int dm = M \quad \text{(mass of the body)}$$

Centre of Mass of Some Common Systems:

• A system of two point masses: $$m_1 r_1 = m_2 r_2$$

• Rectangular plate (By symmetry): $x_c = \frac{b}{2}, \quad y_c = \frac{L}{2}$

• A triangular plate (By qualitative argument)

• A semi-circular ring

• A semi-circular disc: $$y_c = \frac{4R}{3\pi}, \quad x_c = 0$$

• A hemispherical shell: $$y_c = \frac{R}{2}, \quad x_c = 0$$

• A solid hemisphere: $$y_c = \frac{3R}{8}, \quad x_c = 0$$

• A circular cone (solid): $$y_c = \frac{h}{4}$$

• A circular cone (hollow): $$y_c = \frac{h}{3}$$

Motion of Centre of Mass and Conservation of Momentum:

• Velocity of Centre of Mass of System:

$$\vec{V} _{cm} = \frac{m _1\vec{V} _1 + m _2\vec{V} _2 + m _3\vec{V} _3 + \ldots + m _n\vec{V} _n}{M}$$

$$\vec{P} _{\text{System}} = M \vec{V} _{cm}$$

• Acceleration of Centre of Mass of System:

$$\vec{a} _{cm} = \frac{m _1\vec{a} _1 + m _2\vec{a} _2 + m _3\vec{a} _3 + \ldots + m _n\vec{a} _n}{M}$$

$$\vec{a} _{cm} = \frac{\text{Net External Force}}{M}$$

$$\vec{F} _{\text{ext}} = M\vec{a} _{cm}$$

Rotational Kinetic Energy:

Rotational kinetic energy $K_{\text{rot}}$:

$$K_{\text{rot}} = \frac{1}{2} I \omega^2$$

where $I$ represents the moment of inertia of the object about the axis of rotation, and $\omega$ is the angular velocity.