System of Particles & COM

Cheatsheet Content

Rigid Bodies and Systems of Particles Rigid Body: An object where the distance between any two constituent particles remains constant. Shape doesn't change. System of Particles: An extended object where particles can move relative to each other. Motion: Translational: All particles have the same velocity. Rotational: Particles have different velocities, undergoing circular motion about an axis. Combined (Translational + Rotational) motion is possible. Center of Mass (COM) The point where the entire mass of the system is assumed to be concentrated for studying translational motion. Discrete Particles For 2 particles ($m_1, m_2$) along the x-axis: $$ X_{com} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} $$ If $m_1 = m_2 = m$: $$ X_{com} = \frac{m x_1 + m x_2}{m + m} = \frac{x_1 + x_2}{2} $$ For 3D space ($m_1, m_2, ..., m_n$): $$ X_{com} = \frac{\sum m_i x_i}{\sum m_i} $$ $$ Y_{com} = \frac{\sum m_i y_i}{\sum m_i} $$ $$ Z_{com} = \frac{\sum m_i z_i}{\sum m_i} $$ In Vector form: $$ \vec{R}_{com} = \frac{\sum m_i \vec{r}_i}{\sum m_i} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + ... + m_n \vec{r}_n}{M_{total}} $$ where $\vec{r}_i = x_i \hat{i} + y_i \hat{j} + z_i \hat{k}$ is the position vector of the $i^{th}$ particle. Continuous Bodies For continuous mass distribution, replace summation with integration: $$ X_{com} = \frac{1}{M} \int x \, dm $$ $$ Y_{com} = \frac{1}{M} \int y \, dm $$ $$ Z_{com} = \frac{1}{M} \int z \, dm $$ where $dm$ is a small mass element and $M = \int dm$ is the total mass. Mass per unit length ($\lambda$): $\lambda = \frac{M}{L}$ (for 1D objects like rods) Then $dm = \lambda \, dx = \frac{M}{L} dx$. Mass per unit area ($\sigma$): $\sigma = \frac{M}{A}$ (for 2D objects like plates) Then $dm = \sigma \, dA = \frac{M}{A} dA$. Mass per unit volume ($\rho$): $\rho = \frac{M}{V}$ (for 3D objects) Then $dm = \rho \, dV = \frac{M}{V} dV$. COM of Uniform Geometrical Shapes For uniform density, COM coincides with the geometric center. Uniform Rod: At its midpoint. Uniform Ring: At its center. Uniform Disc: At its center. Uniform Solid Sphere/Hollow Sphere: At its center. Uniform Cube/Cuboid: At its geometric center. Uniform Triangle: At the intersection of its medians (centroid). If vertices are $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, then: $$ X_{com} = \frac{x_1 + x_2 + x_3}{3}, \quad Y_{com} = \frac{y_1 + y_2 + y_3}{3} $$ Uniform Semicircular Ring of Radius R: $Y_{com} = \frac{2R}{\pi}$ (from the diameter). $X_{com}=0$. Uniform Semicircular Disc of Radius R: $Y_{com} = \frac{4R}{3\pi}$ (from the diameter). $X_{com}=0$. Uniform Solid Hemisphere of Radius R: $Y_{com} = \frac{3R}{8}$ (from the flat base). $X_{com}=0, Z_{com}=0$. Uniform Hollow Hemisphere of Radius R: $Y_{com} = \frac{R}{2}$ (from the flat base). $X_{com}=0, Z_{com}=0$. Uniform Cone (Solid): $Y_{com} = \frac{h}{4}$ (from the base). Uniform Cone (Hollow): $Y_{com} = \frac{h}{3}$ (from the base). Motion of Center of Mass Velocity of COM: $$ \vec{V}_{com} = \frac{\sum m_i \vec{v}_i}{\sum m_i} = \frac{m_1 \vec{v}_1 + m_2 \vec{v}_2 + ... + m_n \vec{v}_n}{M_{total}} $$ Momentum of COM: $$ \vec{P}_{com} = M_{total} \vec{V}_{com} = \sum m_i \vec{v}_i = \sum \vec{p}_i $$ The total momentum of a system of particles is equal to the momentum of its center of mass. Acceleration of COM: $$ \vec{A}_{com} = \frac{\sum m_i \vec{a}_i}{\sum m_i} = \frac{\vec{F}_{ext}}{M_{total}} $$ where $\vec{F}_{ext}$ is the net external force acting on the system. If $\vec{F}_{ext} = 0$, then $\vec{A}_{com} = 0$, which implies $\vec{V}_{com}$ is constant. The COM moves with constant velocity (or remains at rest). Internal forces do not affect the motion of the center of mass. Linear Momentum and Impulse Linear Momentum: $\vec{p} = m\vec{v}$ (where $m$ is mass, $\vec{v}$ is velocity). Newton's Second Law: $\vec{F} = \frac{d\vec{p}}{dt}$. If $m$ is constant, $\vec{F} = m\vec{a}$. Impulse: $\vec{J} = \int \vec{F} \, dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$. Impulse is the change in momentum. Impulse-Momentum Theorem: The impulse applied to an object is equal to the change in its momentum. Conservation Laws Conservation of Linear Momentum: If the net external force on a system is zero ($\vec{F}_{ext} = 0$), then the total linear momentum of the system remains constant ($\vec{P}_{total} = \text{constant}$). $$ m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{v}_1' + m_2\vec{v}_2' $$ (for two particles) Conservation of Energy: Energy can neither be created nor destroyed; it can only be transformed from one form to another. For a conservative system, $KE_i + PE_i = KE_f + PE_f$. Collisions Interactions between objects where forces act over a short time, leading to a change in momentum. Types of Collisions: Elastic Collision: Both momentum and kinetic energy are conserved. The objects separate after collision. For 1D elastic collision: Conservation of momentum: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$ Conservation of KE: $\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$ Relative velocity of approach = Relative velocity of separation: $u_1 - u_2 = -(v_1 - v_2) = v_2 - v_1$ Velocities after collision: $$ v_1 = \frac{(m_1 - m_2)u_1 + 2m_2 u_2}{m_1 + m_2} $$ $$ v_2 = \frac{(m_2 - m_1)u_2 + 2m_1 u_1}{m_1 + m_2} $$ Inelastic Collision: Momentum is conserved, but kinetic energy is NOT conserved (some KE is lost as heat, sound, deformation). The objects may or may not separate. Perfectly Inelastic Collision: Momentum is conserved. Kinetic energy is NOT conserved. The objects stick together after collision and move with a common final velocity ($v_1 = v_2 = v$). $$ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v $$ $$ v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} $$ Coefficient of Restitution ($e$): $$ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} = - \frac{v_2 - v_1}{u_2 - u_1} = \frac{v_1 - v_2}{u_2 - u_1} $$ For Elastic Collision: $e=1$ For Perfectly Inelastic Collision: $e=0$ For Inelastic Collision: $0 Rotation Angular Displacement: $\Delta \theta$ (in radians) Angular Velocity: $\omega = \frac{d\theta}{dt}$ (in rad/s) Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ (in rad/s$^2$) Relationship between Linear and Angular Quantities: Arc Length: $s = r\theta$ Linear Velocity: $v = r\omega$ (for a point at distance $r$ from axis of rotation) Linear Acceleration: $a_t = r\alpha$ (tangential acceleration) Torque ($\vec{\tau}$) The rotational equivalent of force. It causes angular acceleration. Definition: $\vec{\tau} = \vec{r} \times \vec{F}$ Magnitude: $\tau = r F \sin\theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$. Alternatively, $\tau = r_{\perp} F = r F_{\perp}$, where $r_{\perp}$ is the perpendicular distance from the pivot to the line of action of the force (moment arm), and $F_{\perp}$ is the component of force perpendicular to $\vec{r}$. Units: Newton-meter (N·m) Newton's Second Law for Rotation: $\vec{\tau}_{net} = I \vec{\alpha}$ where $I$ is the moment of inertia and $\vec{\alpha}$ is the angular acceleration. Condition for Rotational Equilibrium: $\sum \vec{\tau} = 0$