91Ó°ÊÓ

Understanding the total mass of a system is a fundamental concept in physics, serving as a base for various other calculations. When dealing with a system of particles, the total mass is simply the sum of the masses of individual particles.

In our textbook exercise, the total mass is found by utilizing the y-coordinate equation for the center of mass, which is \(y_{cm} = \frac{\sum m_i y_i}{\sum m_i}\). This equation essentially states that the center of mass is the weighted average position of all the masses in the system. Here, one particle is at the origin with an unknown mass \(m_1\), and the other particle has a known mass of \(0.50 \text{kg}\) and is located at \(y=6,0 \text{m}\).

Given the center of mass is at \(y=2.4 \text{m}\), we set up the equation \(2.4m = \frac{m_1\cdot0 + 0.50\text{kg}\cdot6m}{m_1 + 0.50\text{kg}}\), which upon solving gives us the value of \(m_1\). The sum of \(m_1\) and \(0.50 \text{kg}\) yields the total mass of the system. Knowing this allows us to delve deeper into understanding the system's dynamics, especially when observing how external forces interact with the mass.

The acceleration of the center of mass is pivotal in describing how the velocity of the system's mass center changes over time. In this exercise, the velocity of the center of mass is provided as a function of time: \(v = 0.75t^2\).

To find the acceleration at any point in time, we look at the rate of change of velocity, which is the derivative of the velocity function with respect to time. By differentiating the velocity \(v\) with respect to time \(t\), we obtain the acceleration equation \(a = \frac{dv}{dt} = 1.5t\). This simple relation shows that the acceleration of the system's center of mass is directly proportional to time and will help us later in calculating the net external force acting on the system.

The net external force is a concept that emerges from Newton's second law of motion, which states that force is the product of mass and acceleration \(F = ma\). This law implies that the total force acting on an object or system is the cause of its acceleration.

In order to find the net external force acting on the system at a particular time, we need both the total mass calculated previously and the acceleration of the center of mass at that instant. After evaluating the expression for acceleration derived from the velocity—\(a = 1.5t\)—at \(t = 3.0 \text{s}\), we plug this value into the equation for force, along with the total mass of the system. This calculation deciphers the net external force acting on the system at the specified time, providing insight into the influence of the surroundings on the system's motion.