Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Speed and collisions ↓


Center of mass and speed


In classical mechanics, the concept of the center of mass and its motion is important for understanding the dynamics of a system, especially when examining motion and collisions. By definition, the center of mass is the point at which the entire mass of a system can be considered concentrated for the purpose of analysis. Understanding this concept allows us to simplify complex problems by reducing them into more manageable parts. Let's start by defining what the center of mass is and how it is calculated in different scenarios.

Understanding the center of mass

The center of mass of an object or system of particles is the point where the weighted relative position of the distributed mass is zero. In simple terms, it is the average location of the entire mass of the object. For a single object with uniform density, the center of mass will be at its geometric center. However, for systems of objects or objects with non-uniform density, the calculation can be more complicated.

Center of mass of a single object

For a single symmetrical object such as a sphere or cube, the center of mass is at the center of the object. Consider a simple object such as a ruler. Assuming the ruler has uniform density, the center of mass is exactly at its midpoint.

Center of mass of many objects

The center of mass for a system consisting of many particles can be calculated using the following formula:

        R_cm = (Σ m_i * r_i) / Σ m_i

Here, R_cm is the position vector of the center of mass, m_i is the mass of the i-th particle, and r_i is the position vector of the i-th particle.

For example, consider two particles of masses 2 kg and 3 kg placed at positions (1,0) and (4,0) on the x-axis, respectively. The center of mass R_cm will be:

        R_cm = [(2 * 1) + (3 * 4)] / (2 + 3) = (2 + 12) / 5 = 14/5 = 2.8

This means that the center of mass is located on the x-axis at x = 2.8.

Speed of center of mass

When discussing motion in classical mechanics, the trajectory of the center of mass is often considered. The center of mass moves as if all the mass and external forces of the system were concentrated at this point.

The important point is that the centre of mass of an isolated system (i.e., there are no external forces acting on the system) remains in uniform motion or at rest. This principle is known as conservation of momentum. Let us consider some scenarios to see how this principle applies.

Example of motion of the center of mass

Imagine two ice skaters standing still on a frictionless ice surface and pushing each other. Skater A has a mass of 50 kg, and skater B has a mass of 70 kg. After the push, skater A moves to the right at a velocity of 2 m/s, and skater B moves to the left. Skater B's velocity can be calculated using conservation of momentum:

        Initial momentum = Final momentum 0 = (50 * 2) + (70 * v) 0 = 100 + 70v 70v = -100 v = -100/70 v ≈ -1.43 m/s

Skater B will move to the left at a velocity of approximately -1.43 m/s.

Collision and center of mass

Collisions are events where two or more bodies exert forces on each other in approximately the same amount of time. In physics, collisions can be classified as elastic or inelastic based on the conservation of kinetic energy.

Elastic collision

In an elastic collision both momentum and kinetic energy are conserved. An example of an elastic collision is when two identical billiard balls collide with each other.

Suppose a ball with a mass of 1 kg collides with another stationary ball moving at a speed of 3 m/s. After the collision, the second ball moves at a speed of 3 m/s, and the first ball stops.

        Momentum before = Momentum after (1 * 3) + (1 * 0) = (1 * 0) + (1 * 3) 3 = 3

In this collision both momentum and kinetic energy are conserved.

Inelastic collision

In an inelastic collision, momentum is conserved, but kinetic energy is not necessarily conserved. Some or all of the kinetic energy is converted into other energy forms, such as heat or sound.

Suppose two cars collide and stick together after the collision. This is a perfectly inelastic collision where maximum kinetic energy is transformed.

Example of center of mass in collisions

Consider a system of two colliding balls. The center of mass of this system will continue to move with the same velocity, provided there are no external forces.

Practical applications of center of mass

Understanding the center of mass is valuable in many fields. In sports, athletes can adjust their center of mass to maintain balance or increase speed. Engineers use these principles when designing stable structures and vehicles.

For example, when designing a car, engineers make sure that the engine, passengers and luggage are balanced with the car's center of mass, so that the car can avoid tipping over during sharp turns.

Conclusion

The concepts of the center of mass and its motion provide essential information for analyzing motion and collisions in classical mechanics. Whether calculating the center of mass of a system or predicting the outcomes of collisions, these concepts help simplify and solve complex problems.

Particle A Particle B

In the above example, two particles on a line, red and blue, depict a simple system for analyzing their center of mass and the resulting motion. With these principles, one can predict, design, and explain the results observed in many systems in the real world.


Undergraduate → 1.4.5


U
username
0%
completed in Undergraduate

← Prev (1.4.4)
Elastic and inelastic collision
Next (1.4.6) →
Rocket Propulsion

Comments 



Center of mass and speed

https://www.physicsflow.com/ug/1.4.5?pfal=en