91Ó°ÊÓ

Understanding mass and velocity vectors is crucial when calculating the center of mass velocity. Imagine each object in a system as having its own mass, which is a measure of the amount of matter it contains. The velocity vector represents both the speed of the object and its direction.
For example, the velocity vector \(2 \hat{i} - 7 \hat{j} + 3 \hat{k}\) indicates a velocity with components in the direction of the coordinate axes, with 2 units along the \(i\)-axis, \(-7\) units along the \(j\)-axis, and 3 units along the \(k\)-axis.
In a system of particles, each particle's velocity vector is adjusted by its mass, resulting in a new vector: the mass times velocity vector. This vector represents the momentum of the particle and is essential in finding the overall motion of the system.

A system of particles is essentially a group of masses that can interact with one another. These masses could be anything from planets in a galaxy to atoms in a gas. In our exercise, we are dealing with two particles: one with a mass of 10 kg and another with 2 kg.
Each of these particles not only moves with its specific velocity vector but also contributes differently to the motion of the entire system based on its mass.

• Particle 1: Mass = 10 kg, Velocity = \(2 \hat{i} - 7 \hat{j} + 3 \hat{k}\)
• Particle 2: Mass = 2 kg, Velocity = \(-10 \hat{i} + 35 \hat{j} - 3 \hat{k}\)

The center of mass of the system is influenced more by the more massive particle, but the overall movement is the combined effect of all particles. Understanding this helps in calculating where the overall mass balance lies and predicting how the system behaves as a whole.

The calculation of the center of mass involves finding a point that acts as an average position of all the masses in the system. For this, we use the formula for velocity of the center of mass: \( \vec{v}_{\text{cm}} = \frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2} \). This formula helps us in combining the effects of all particles based on their mass and velocity.
When calculating, start by multiplying the mass of each particle by its velocity vector. For instance, with our exercise, find \( m_1 \vec{v}_1 = 20 \hat{i} - 70 \hat{j} + 30 \hat{k} \) and \( m_2 \vec{v}_2 = -20 \hat{i} + 70 \hat{j} - 6 \hat{k}\).
Next, add these results to find the total mass-weighted velocity: \(20 \hat{i} - 70 \hat{j} + 30 \hat{k} + (-20 \hat{i} + 70 \hat{j} - 6 \hat{k}) = 0 \hat{i} + 0 \hat{j} + 24 \hat{k}\).
Finally, divide by the total mass of the system to find the velocity of the center of mass: \( \vec{v}_{cm} = \frac{24 \hat{k}}{12} = 2 \hat{k} \). This result demonstrates that the system's center of mass moves entirely in the \( k \)-direction.