91Ó°ÊÓ

In physics, a **system of particles** refers to a group of particles that are being analyzed as a single entity. Each particle has its own mass and position within the system, and together, they create the entire system's dynamics. When working with such a system, one crucial concept is the **center of mass**.In your exercise, you're dealing with two particles. The first particle has a mass of **0.50 kg** and is located at **y = 6.0 m** on the y-axis. The second particle, whose mass we seek, is at the origin. The position of the center of mass is given by\[ y_{cm} = \frac{m_1y_1 + m_2y_2}{m_1 + m_2}, \]where \( m_1 = 0.50 \text{ kg} \) and \( y_1 = 6.0 \text{ m} \).- **Center of Mass Position:** Since the center of mass is located at **y = 2.4 m** along the y-axis, you can use this information to calculate the unknown mass \( m_2 \). The particle at the origin contributes \( y_2 = 0 \), simplifying the calculation.- **Total Mass Calculation:** Adding the masses of all particles gives the system's total mass. This information is used later to find other properties like force and acceleration.

**Velocity** refers to the speed at which something is moving in a certain direction. In this exercise, the velocity of the center of mass of the system of particles is given by the formula\[ v_{cm} = (0.75 \text{ m/s}^3) \cdot t^2. \]This expression means that the velocity of the center of mass increases as time progresses, and this change is dependent on time squared, indicating an accelerated motion.**Acceleration** is the rate at which velocity changes over time. You find the acceleration of the center of mass by taking the time derivative of velocity:\[ a_{cm} = \frac{dv_{cm}}{dt} = \frac{d}{dt}(0.75 \cdot t^2) = 1.5 \cdot t. \]- **Understanding Acceleration:** This result means that for every unit time \( t \), the center of mass's velocity increases by \( 1.5 \text{ m/s}^2 \).Tracking velocity and acceleration in these calculations helps us understand how the system's particles will move collectively over time, which is essential for predicting future states of motion.

The **net external force** acting on a system is crucial for understanding how the system moves as a whole. According to Newton's second law of motion, force is the product of mass and acceleration:\[ F = m_{total} \cdot a_{cm}. \]In this exercise, you've already calculated the total mass of the system \( m_{total} = 1.25 \text{ kg} \) and its acceleration \( a_{cm} = 1.5t \text{ m/s}^2 \). At a specific time, such as \( t = 3.0 \text{ s} \), the acceleration can be calculated by substituting \( t \) into the equation:\[ a_{cm} = 1.5 \cdot 3.0 = 4.5 \text{ m/s}^2. \]Using these values, the net external force is then calculated as:\[ F = 1.25 \cdot 4.5 = 5.625 \text{ N}. \]- **Application of Force**: Understanding the net force helps predict how the system accelerates and changes its state of motion due to external influences. This allows for anticipation of the dynamic behaviors of the system over time. Grasping how external forces interact with systems of particles enables you to analyze real-world movements and behaviors systematically.