The system of three particles has the indicated particle masses, velocities, and external forces

The system of three particles has the indicated particle...

Key Concepts

Center of Mass

The center of mass is a weighted average position of all the particles in a system, taking their masses into account. It is defined as the sum of each particle's mass multiplied by its position vector divided by the total mass of the system. This concept allows us to simplify a multi-particle system into a single point where the total mass is assumed to be concentrated, which is particularly useful in analyzing overall system motion in mechanics.

Velocity and Acceleration of the Center of Mass

The velocity of the center of mass is found by taking the time derivative of the center of mass position, and similarly, the acceleration is the second derivative with respect to time. These derivatives describe how the overall position of the system changes over time and are central to applying Newton's laws to the system as a whole, especially under the influence of external forces.

Kinetic Energy of a Particle System

Kinetic energy in a system of particles is the sum of the individual kinetic energies of the particles. It can also be expressed as a combination of the kinetic energy associated with the center of mass motion and the kinetic energy associated with motion relative to the center of mass. This decomposition is useful for analyzing energy in the system and for understanding how energy is distributed between overall motion and internal dynamics.

Total Linear Momentum

The total linear momentum of a system is the vector sum of the momenta of the individual particles, with each particle’s momentum defined as the product of its mass and velocity. This concept is crucial because, in the absence of external forces, the total linear momentum of the system remains conserved, providing a powerful tool for analyzing collisions and system dynamics.

Angular Momentum of a System

Angular momentum of a system is calculated by taking the cross product of each particle's position vector with its momentum and then summing these products over all particles. It provides a measure of the rotational motion of the system about a point (often the origin or the center of mass) and is conserved in the absence of external torques, making it a key concept in the study of rotational dynamics.

Transcript

00:01 In this problem, on the topic of kinetics of systems of particles, we are given a system of three particles with the masses given as well as the velocities and external forces.

00:12 We want to find the mass center, the first and second derivative of the mass center, the kinetic energy of the system t, the angular momentum, and the first derivative with respect to time of this angular momentum for this two -dimensional system.

00:28 So firstly to find the mass center, which is r, we know that this mass center r is defined as the sum of the product of the masses times their positions divided by the sum of all their masses.

00:53 And so this is m times a distance of d along unit vector i plus 2m.

01:07 Into 2d times j plus 4m times 1 .5d times k.

01:19 All of this divided by the sum of the masses m plus 2m plus 4m.

01:30 So simplifying, we get this to be d over 7 into unit vector i plus 4x4.

01:45 Plus 6 times unit vector k.

01:49 And so that's the position of the mass center of the system of particles.

01:57 Next we want to find the first derivative of this mass center, r dot, and this is defined as the sum of mi, r i.

02:16 R -i dot over the sum of the masses.

02:20 So this is equal to m into 2v j plus 2m into 3m into 3v k, where v is the velocity of the system plus 4m into v i, all divided by 7m.

03:05 And so calculating this becomes v over 7.

03:11 So this is essentially the velocity of the mass center of the system into 4i plus 2j plus 6k...