Rotational Motion: Centre of Mass

Vocabulary and key formulas covering standard centres of mass, discrete particle coordinate calculations, center of mass kinematics, conservation of motion, and composite/truncated body formulas.

Centre of Mass of a Thin Uniform Straight Rod

Located at the geometric centre of the rod.

Centre of Mass of a Triangular Plate

Located at the centroid of the triangular plate.

Centre of Mass of a Hollow Cone

Located at a height of $\frac{h}{3}$ from the base along the central axis.

Centre of Mass of a Solid Cone or Pyramid

Located at a height of $\frac{h}{4}$ from the base along the central axis.

Centre of Mass of a Semi-Circular Arc

Located at a distance of $\frac{2R}{\pi}$ from its centre on the axis of symmetry.

Centre of Mass of a Semi-Circular Disc

Located at a distance of $\frac{4R}{3\pi}$ from its centre $O$ on the axis of symmetry.

Centre of Mass of a Solid Hemisphere

Located at a distance of $\frac{3R}{8}$ from its centre $O$ on the axis of symmetry.

Centre of Mass of a Hollow Hemisphere (Hemispherical Shell)

Located at a distance of $\frac{R}{2}$ from its centre $O$ on the axis of symmetry.

Centre of Mass of a Horse-Shoe Magnet

Located at its geometric centre, outside the material of the body but within the boundary limits.

Position Vector of the Centre of Mass ($\vec{r}_{\text{cm}}$)

For a system of particles with masses $m_1, m_2, \dots, m_n$ and position vectors $\vec{r}_1, \vec{r}_2, \dots, \vec{r}_n$, it is defined as $\vec{r}_{\text{cm}} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$.

Condition for Zero Centre of Mass Displacement

If the net external force on a system is zero ($F_{\text{ext}} = 0 \Rightarrow a_{\text{cm}} = 0$) and the system is initially at rest ($u_{\text{cm}} = 0$), then $v_{\text{cm}} = 0$ and the displacement of the centre of mass is zero ($S_{\text{cm}} = 0$).

Displacement Relation for Unaltered Centre of Mass

Given by the equation $m_1 d_1 + m_2 d_2 + m_3 d_3 = 0$, where $d$ represents the directional displacement of each mass along an axis when the position of the centre of mass does not change.

Meeting Point of Mutually Interacting Particles

When two isolated particles approach each other solely due to their mutual internal interaction, they always meet at their centre of mass.

Boat Problem: Boat Displacement

The distance moved by a boat of mass $M$ in the opposite direction when a boy of mass $m$ walks from one end of the boat of length $l$ to the other: $\Delta x_{\text{boat}} = \frac{m l}{M + m}$.

Boat Problem: Boy Displacement with Respect to Ground

The net distance traversed by the boy relative to the ground when walking across the boat of length $l$: $\Delta x_{\text{boy}} = \frac{M l}{M + m}$.

Boat Problem: Velocity of Boat with Respect to Ground ($V$)

Given the boy walks with velocity $V_r$ relative to the boat, the ground-frame velocity of the boat is $V = -\frac{m V_r}{M + m}$, where the negative sign denotes motion opposite to the boy.

Boat Problem: Velocity of Boy with Respect to Ground ($V'$)

Given the boy walks with velocity $V_r$ relative to the boat of mass $M$, the boy's velocity relative to the ground is $V' = \frac{M V_r}{M + m}$.

Centre of Mass of Truncated (Cut-out) Bodies

Calculated by treating the removed portion as a negative mass using the formula $r_{\text{cm}} = \frac{m_1 r_1 - m_2 r_2}{m_1 - m_2}$, where subscript 1 refers to the original complete body and subscript 2 refers to the cut-out section.

Mass Proportionality Substitutions for Non-Uniform/Uniform Objects

When masses are not explicitly provided for uniform density objects, length is substituted for 1D bodies, area is substituted for 2D bodies, and volume is substituted for 3D bodies.

Shift of Mass in a Two-Particle System

If mass $m_1$ is shifted towards $m_2$ by a distance $d$, the distance $m_2$ must be shifted towards $m_1$ to keep the centre of mass fixed is $d' = \frac{m_1 d}{m_2}$.