91Ó°ÊÓ

Newton's Second Law is a fundamental principle in physics that relates the force exerted on an object to its acceleration. The law is succinctly expressed in the formula \( F = ma \), where \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is its acceleration. In this exercise, we are looking at a system of particles subjected to known forces. By applying Newton's Second Law, we can determine how these forces cause the particles to accelerate.
For instance, when a force \( \vec{F}_1 = (-4.00 \hat{\mathrm{i}} + 5.00 \hat{\mathrm{j}}) \text{ N} \) acts on a particle with mass \( 2.00 \times 10^{-3} \text{ kg} \), we find its acceleration by rearranging our equation: \( a_1 = \frac{F_1}{m_1} \).
This results in an acceleration \( \vec{a}_1 = (-2000 \hat{\mathrm{i}} + 2500 \hat{\mathrm{j}}) \text{ m/s}^2 \) for the first particle. Understanding this relationship supports the analysis of motion, explaining how varying forces influence particle behavior.

The center of mass of a system of particles is a point that behaves as if all the system's mass and external forces are concentrated there. It's a crucial concept for solving dynamics problems involving multiple particles. The position of the center of mass \( \vec{s}_{cm} \) can be determined by weighing individual positions by their respective masses and then summing and normalizing across the total mass.
In the given problem, we calculate the displacement of the center of mass, \( \vec{s}_{cm} \), as follows: \( \vec{s}_{cm} = \frac{m_1 \vec{s}_1 + m_2 \vec{s}_2}{m_1 + m_2} \).
This formula tells us how the entire system shifts based on individual particle displacements. By plugging in the known values, we determine the center of mass displacement as \((-0.667 \hat{\mathrm{i}} + 0.333 \hat{\mathrm{j}}) \times 10^{-3} \text{ m} \). This displacement helps analyze how the system's orientation changes under applied forces across a short period (\(t = 2.00 \text{ ms}\)).

Kinetic Energy (KE) quantifies the motion of an object or system of particles. It's calculated using the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity. For a system of particles, we often consider the kinetic energy of the center of mass, which represents the overall movement of the system.
To find the kinetic energy of the center of mass at \( t = 2.00 \text{ ms} \), we first compute the center of mass velocity \( \vec{v}_{cm} \) resulting from applied forces. The velocity can be found from: \( \vec{v}_{cm} = \frac{m_1 \vec{a}_1 + m_2 \vec{a}_2}{m_1 + m_2} \times t \).
Putting values in the equation provides us with \( \vec{v}_{cm} = (-2.67 \hat{\mathrm{i}} + 1.33 \hat{\mathrm{j}}) \text{ m/s} \). Subsequently, the kinetic energy is calculated: \[ KE_{cm} = \frac{1}{2}M_{total}(\sqrt{(-2.67)^2 + 1.33^2})^2 \].
This results in a kinetic energy of \( 0.0267 \text{ J} \), offering insight into the dynamic behavior of the system influenced by the particles' combined forces and mass.