Centre Of Mass
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General Learning Resources
Mass Moment and Centre of Mass:
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Mass Moment: $\vec{M} = m \vec{r}$
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Centre of Mass of a System of ‘N’ Discrete Particles: $\vec{r}_{cm} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + \ldots + m_n\vec{r}_n}{m_1 + m_2 + \ldots + m_n}$
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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.
BETA
