COM Physics PDF

## Center of Mass ### Study of Kinematics vs. Kinetics * Kinematics: Study of motion without considering forces and energy. * Kinetics: Study of the effects of forces and energy on motion. * Includes: Newton's Laws of motion, work-energy methods, impulse and momentum methods. * The methods for work and energy and impulse and momentum methods are developed using the equation F = mā. * These methods are advantageous because they do not rely on determining acceleration. * Work and energy methods relate force, mass, velocity, and displacement. * Impulse and momentum methods relate force, mass, and the time interval of action. * The impulse and momentum principle are developed from Newton's second law. * The work-energy theorem and impulse-momentum methods can be applied to a single particle (rigid body) in translational motion. ### System of Particles * A **system of particles** is a defined collection of particles. These particles may or may not interact and may be actual particles or parts of rigid bodies. * **Internal forces** act between particles within the system. * These forces always come in equal magnitude and opposite direction pairs. * **External forces** act on particles from bodies outside the system. * **Extended bodies** are systems containing an infinitely large number of particles with infinitely small separations between them. * A **deformable body** is a body where the particle separations and relative locations change. * A **rigid body** is a body in which particle separations and relative locations remain unchanged. #### Mass Center * Regardless of the complexity of a system of particles, there is a **mass center** (CM) present. * The CM is a special point whose translation motion is characteristic of the system. * All the mass of a rigid body or system of particles can be conceptually concentrated at the CM. It is located closer to more massive particles. * If a force is applied to an extended body at its CM, only translational motion will occur. If the force is applied at another point, the system translates and rotates. ### Center of Mass of a System of Discrete Particles * A **system of discrete particles** is a system of particles with finite distances between them. * Assume n discrete particles with masses: *m*1, *m*2...*m**i*...*m**n* moving with velocities: *v*1, *v*2...*v**i*...*v**n* located at positions: *r*1, *r*2...*r**i*...*r**n*. * The **center of mass** is located at *r*: * *r* = (*m*1*r*1 + *m*2*r*2 + ... + *m**n**r**n* )/ (*m*1 + *m*2 + ... + *m**n*) * *r* = Σ*m**i**r**i* / Σ*m**i* * *r* = Σ*m**i**r**i* / *M* * Where M = Σ*m**i* is the total mass of the system. * The Cartesian coordinates of the center of mass are given by: * *x* = Σ*m**i**x**i* / *M* * *y* = Σ*m**i**y**i* / *M* * *z* = Σ*m**i**z**i* / *M* ### Center of Mass of a System of Two Particles * The center of mass of a two-particle system lies on the line connecting the two particles. * It divides the distance between the two particles in the inverse ratio of their masses. So, if the two masses are the same, the CM is at the midpoint. ### Center of Mass of an Extended Body * An **extended body** is a continuous distribution of mass. * The CM of an extended body can be calculated by: * *r* = (∫ *r* *dm* ) / ∫ *dm* * *r* = (∫ *r* *dm* ) / *M* * Where M is the total mass of the body. * The center of mass of a uniform symmetrical body lies on the axis of symmetry. ### Motion of the Center of Mass * The motion of a system of particles can be considered as the superposition of the translation of the CM and the motion of the particles relative to the CM. * The total linear momentum of a system is equal to the linear momentum of the system due to translation of its CM. * The concept of the CM is useful for analyzing the gross translation of a system. ### Center of Mass Frame of Reference * The **centroidal frame of reference** is the frame fixed to the CM of the system. This frame moves with the CM. * In this frame, the position, velocity, and acceleration of the CM are zero. * The sum of mass moments in this frame vanishes. ### Impulse-Momentum Principle * **Linear impulse (Imp)** is the integral of force over time: * Imp = ∫ *F* *dt* * For a constant force: Imp = *F* Δ*t* * **Linear momentum (p)** is the product of mass and velocity: * *p* = *mv* * **Impulse-Momentum Principle:** The change in momentum of a body is equal to the impulse of the net force acting on it during the given time interval. * Imp = Δ*p* = *p*2 - *p*1 * Where *p*2 is the momentum at time t2 and *p*1 is the momentum at time t1. * **Conservation of Linear Momentum:** If the net impulse of external forces on a system is zero over a time interval, the total linear momentum of the system remains constant. ### Impulsive Motion * An **impulsive force** is a large force that acts for a very short time interval. * **Impulsive motion** refers to the motion of a body under the action of an impulsive force. * External forces that are negligible compared to the impulsive force are considered **non-impulsive**. * When analyzing impulsive motion, non-impulsive forces are usually neglected. ### Motion of the Center of Mass in One Dimension * Consider a system of two bodies with different initial velocities. * The velocity of the CM is the sum of the product of each body's mass and velocity divided by the total mass. * The velocities of each body in the centroidal frame are equal to each body's initial velocity in the inertial frame minus the velocity of the CM. * **Centroidal Frame:** Frame of reference fixed to the CM. ### Motion of the Center of Mass in Vector Form * The velocity of the CM of a system of particles with different velocities is calculated by using the vector sum of the masses and velocities. * The velocity of each particle in the centroidal frame can be found by subtracting the velocity of the CM from the velocity of each particle in the inertial frame. ### Conservation of Linear Momentum for a System of Particles * If the internal forces within a system are zero, the net external force acting on the system is equal to the mass of the system multiplied by its acceleration. * For a system of particles, Newton's 2nd Law can be written as: * Σ *F**i* + Σ *f**i* = m*i* *a* * Σ *f**i* = 0, so Σ *F**i* = M *a* * Where: * Σ *F**i* is the sum of the external forces acting on the particles. * Σ *f**i* is the sum of the internal forces acting on the particles. * *M* is the total mass of the system. * The center of mass of a system accelerates as if it were a single particle with a mass equal to the total mass of the system and subjected to the net external force. ### Application of Newton's Laws of Motion and Momentum to a System of Particles * Newton's third law, which states that every action has an equal and opposite reaction, applies to systems of particles as well. ### Simple Atwood Machine * This machine consists of two masses connected by a string over a pulley. The acceleration of the CM can be determined by applying the principle of conservation of linear momentum. The tension in the string can be determined using the force equation. ### Systems of Particles with a Constant Force * Consider a system of two blocks connected by an un-stretched spring, with a constant force applied to one block. * The acceleration of the CM is equal to the force divided by the total mass of the system. The extension of the spring can be determined by using the equation of motion for a spring. ### Work-Energy Methods for a System of Particles * The work-energy theorem can be applied to each particle in the system or to the system as a whole. * To calculate the work done by internal forces, the concept of potential energy is useful. * If the total work done by internal forces and external forces is zero, then the mechanical energy of the system is conserved. ### Conservation of Mechanical Energy * The total mechanical energy of a system remains constant if the total work done by internal forces not described by a potential energy equation is zero and no external forces act on the system. The total mechanical energy of a system changes by the work done by the external forces. ### Impulsive Forces * Impulsive forces lead to a change in the momentum of a system. * The principle of conservation of momentum states that the total momentum of a system remains constant when no external impulsive forces act on the system. * The impulse of external forces acting on a system in a given time interval is equal to the change in the momentum of the system. ### Variable Mass Systems * A variable mass system is a system where mass is either added or ejected from the system. * **Thrust force:** The force exerted by a mass being added or ejected from a system. The thrust force is equal to the product of the rate of mass loss or gain and the relative velocity of the mass. * **Rocket Propulsion** * Motion of a rocket can be analyzed by considering the force due to the ejection of the exhaust gases. * The velocity of the rocket can be determined using the principle of conservation of momentum. * The final velocity of a rocket is dependent on the initial velocity, fuel ejection velocity, and mass ratio of the rocket - the final mass divided by the initial mass. ### Oblique Collisions * **Oblique collision:** Collision where the velocity vectors of the two colliding bodies are not along the line of impact. * The impact can be analyzed by resolving the velocities into components along the tangent (t) and normal (n) to the line of impact. * **Conservation of momentum:** The component of momentum along the tangent is conserved. * **Coefficient of restitution:** The coefficient of restitution determines the relative speed of the two bodies after the collision along the normal direction. ### Perfectly Inelastic Collisions * In a **perfectly inelastic collision**, the two bodies stick together after colliding. * The coefficient of restitution for perfectly inelastic collisions is zero. ### Perfectly Elastic Collisions * **Perfectly elastic collision:** Kinetic energy is conserved. * The coefficient of restitution is 1. ### Kinetic Energy in Perfectly Elastic Impact - The kinetic energy of a system of particles is the sum of the kinetic energies of all the particles. - When a collision is perfectly elastic, the kinetic energy of the system before and after the collision is conserved. ### Inelastic Collisions - In an inelastic collision, some of the kinetic energy is lost. - In a perfectly inelastic collision, some of the kinetic energy is lost as heat and sound.

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