System of Particles & COM

Cheatsheet Content

### System of Particles - **Definition:** A collection of a large number of particles which interact with each other. The motion of the system is described by the motion of its center of mass and the motion of particles relative to the center of mass. - **Internal Forces ($F_{ij}$):** Forces exerted by particles within the system on each other. They always occur in action-reaction pairs (Newton's 3rd Law) and sum to zero for the entire system ($\sum \vec{F}_{int} = 0$). - **External Forces ($F_{ext}$):** Forces exerted on the particles of the system by agents outside the system. - **Net Force on System:** $\vec{F}_{net} = \sum \vec{F}_{ext} + \sum \vec{F}_{int} = \sum \vec{F}_{ext}$ (since internal forces cancel out). ### Center of Mass (COM) - **Definition:** A unique point where the entire mass of the system is assumed to be concentrated and all external forces are assumed to act. It's an imaginary point that represents the average position of all the mass in the system. - **Position Vector of COM:** - For discrete particles: $\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + ...}{M}$ - For continuous mass distributions: $\vec{R}_{CM} = \frac{\int \vec{r} dm}{\int dm} = \frac{1}{M} \int \vec{r} dm$ - In Cartesian coordinates for discrete particles: $$X_{CM} = \frac{\sum m_i x_i}{M}$$ $$Y_{CM} = \frac{\sum m_i y_i}{M}$$ $$Z_{CM} = \frac{\sum m_i z_i}{M}$$ - **COM of a Two-Particle System:** If origin is at $m_1$, then $X_{CM} = \frac{m_2 x_2}{m_1 + m_2}$. If COM is origin, then $m_1 r_1 = m_2 r_2$. ### Velocity and Acceleration of COM - **Velocity of COM:** - $\vec{V}_{CM} = \frac{d\vec{R}_{CM}}{dt} = \frac{\sum m_i \vec{v}_i}{M}$ - If $\vec{V}_{CM}$ is constant, then $\sum m_i \vec{v}_i = \text{constant}$ (conservation of total momentum). - **Acceleration of COM:** - $\vec{A}_{CM} = \frac{d\vec{V}_{CM}}{dt} = \frac{\sum m_i \vec{a}_i}{M}$ - From Newton's 2nd Law for the system: $\vec{F}_{ext} = M \vec{A}_{CM}$ - This is the most crucial result: The center of mass of a system of particles moves as if all the mass of the system were concentrated at the center of mass and all external forces were applied at that point. ### Linear Momentum and COM - **Total Linear Momentum of System ($P_{sys}$):** $\vec{P}_{sys} = \sum \vec{p}_i = \sum m_i \vec{v}_i = M \vec{V}_{CM}$ - **Newton's 2nd Law (in terms of momentum):** $\vec{F}_{ext} = \frac{d\vec{P}_{sys}}{dt}$ - **Conservation of Linear Momentum:** If $\vec{F}_{ext} = 0$, then $\frac{d\vec{P}_{sys}}{dt} = 0$, which implies $\vec{P}_{sys} = \text{constant}$. - This means $\vec{V}_{CM} = \text{constant}$. The COM moves with constant velocity (or remains at rest) if no external forces act on the system. - Internal forces do NOT change the total linear momentum of the system. ### Frames of Reference - **Laboratory Frame (L-frame):** An inertial frame where the observer is at rest relative to the ground. - **Center of Mass Frame (CM-frame):** A special inertial frame of reference in which the center of mass is at rest. - In the CM-frame, $\vec{V}_{CM}' = 0$. - The total linear momentum of the system in the CM-frame is always zero: $\vec{P}_{CM}' = \sum m_i \vec{v}_i' = 0$. - The velocity of a particle in the L-frame is $\vec{v}_i = \vec{v}_i' + \vec{V}_{CM}$, where $\vec{v}_i'$ is the velocity in the CM-frame. ### Energy of a System of Particles - **Total Kinetic Energy:** $K_{total} = \frac{1}{2} M V_{CM}^2 + \sum \frac{1}{2} m_i (\vec{v}_i')^2$ - The first term is the kinetic energy of the COM (translational KE of the system). - The second term is the kinetic energy of particles relative to the COM (internal KE, includes rotational and vibrational energy). - **Work-Energy Theorem:** The change in total kinetic energy is equal to the total work done by all forces (internal and external): $\Delta K_{total} = W_{ext} + W_{int}$. - **Special Case for Collisions:** - **Elastic Collision:** Total kinetic energy is conserved. - **Inelastic Collision:** Kinetic energy is NOT conserved (some is converted to heat, sound, deformation). However, total linear momentum is ALWAYS conserved if $\vec{F}_{ext} = 0$. ### Impulse and Collisions - **Impulse ($\vec{J}$):** $\vec{J} = \int \vec{F}_{ext} dt = \Delta \vec{P}_{sys}$ - Impulse is the change in momentum of the system. - **Collision:** A short-duration event involving strong interaction forces between two or more bodies. During a collision, internal forces are typically much larger than external forces, so linear momentum is approximately conserved. - **Coefficient of Restitution ($e$):** - $e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} = \frac{|\vec{v}_{2f} - \vec{v}_{1f}|}{|\vec{v}_{1i} - \vec{v}_{2i}|}$ - For 1D collisions: $e = -\frac{(v_{2f} - v_{1f})}{(v_{2i} - v_{1i})}$ - $e=1$ for perfectly elastic collisions (KE conserved). - $e=0$ for perfectly inelastic collisions (bodies stick together). - $0 ### Variable Mass System (Rocket Propulsion) - **Rocket Equation:** $\text{Thrust } = v_{rel} \frac{dm}{dt}$ - Where $v_{rel}$ is the exhaust velocity of gases relative to the rocket, and $\frac{dm}{dt}$ is the rate of mass ejection. - **Change in velocity of rocket:** $v_f - v_i = v_{rel} \ln\left(\frac{m_i}{m_f}\right) - gt$ - (Ignoring gravity for simplicity, $\Delta v = v_{rel} \ln\left(\frac{m_i}{m_f}\right)$) - **Key Principle:** For a system with variable mass, Newton's 2nd Law needs to be applied carefully to the instantaneous system: $\vec{F}_{ext} = \frac{d\vec{P}}{dt}$. ### Angular Momentum of a System & COM - **Total Angular Momentum:** $\vec{L}_{sys} = \sum \vec{r}_i \times \vec{p}_i$ - **Angular Momentum about COM:** $\vec{L}_{sys} = \vec{R}_{CM} \times \vec{P}_{sys} + \vec{L}_{rel}$ - Where $\vec{L}_{rel}$ is the angular momentum of the particles about the COM. - **Torque and Angular Momentum:** $\vec{\tau}_{ext} = \frac{d\vec{L}_{sys}}{dt}$ - This relation holds true when $\vec{\tau}_{ext}$ is calculated about a fixed point, OR about the COM (even if the COM is accelerating). - **Conservation of Angular Momentum:** If $\vec{\tau}_{ext} = 0$, then $\vec{L}_{sys} = \text{constant}$. - Internal torques do NOT change the total angular momentum of the system. ### Niche Concepts & Tricks - **COM of a System with a Hole:** Treat the hole as a negative mass at its geometric center. - $X_{CM} = \frac{M_{total}X_{total} - M_{hole}X_{hole}}{M_{total} - M_{hole}}$ - **COM for Uniform Objects:** For uniform density, COM coincides with the geometric center. - **Explosion Problems:** If a body explodes, external forces are negligible. The COM continues to move with the same velocity it had just before the explosion. - **Recoil of a Gun:** A classic example of momentum conservation. The momentum of the bullet + gun system is conserved. - **Block-on-Block Problems:** If there's no friction between the lower block and the ground, and friction between the blocks, the COM of the two-block system will not accelerate horizontally. - **Path of COM:** The path of the COM is independent of internal forces. For a projectile breaking up in mid-air, the COM continues along the original parabolic trajectory. - **Zero Momentum Frame (CM-frame):** The total kinetic energy in the CM-frame is $K_{CM}' = \sum \frac{1}{2} m_i (\vec{v}_i')^2$. This is often the minimum kinetic energy of the system. The total kinetic energy in the lab frame is $K_{lab} = K_{CM}' + \frac{1}{2} M V_{CM}^2$.