Problem 11 - Calculus of a Single Variable

Question

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|c|}\hline m_{i} & {12} & {6} & {4.5} & {15} \\\ \hline\left(x_{i}, y_{i}\right) & {(2,3)} & {(-1,5)} & {(6,8)} & {(2,-2)} \\\ \hline\end{array}\)

Step-by-Step Solution

Answer

The center of mass for the given system is at the point (2,2).

Step 1: Calculate the Weighted x-coordinates

First, multiply each mass \( m_i \) by its corresponding x-coordinate \( x_i \). This will give the 'weighted' x-coordinates. The result should be an array like this: \( [12*2, 6*-1, 4.5*6, 15*2] = [24, -6, 27, 30] \)

Step 2: Calculate the Weighted y-coordinates

Next, multiply each mass \( m_i \) by its corresponding y-coordinate \( y_i \) to get the 'weighted' y-coordinates. The result should look like this: \( [12*3, 6*5, 4.5*8, 15*-2] = [36, 30, 36, -30] \).

Step 3: Calculate the Sum of Masses

Add all the masses together to get the total mass of the system. This equals \( 12+6+4.5+15 = 37.5 \) kg.

Step 4: Calculate \((x_{cm}, y_{cm})\)

Find \( x_{cm} \) by dividing the sum of weighted x-coordinates by the total mass. Likewise for \( y_{cm} \). Hence, \( x_{cm} = (24-6+27+30) / 37.5 = 2 \) and \( y_{cm} = (36+30+36-30) / 37.5 = 2 \).

Step 5: Present the Answer

The center of mass for the system of point masses is found to be at the coordinates (2,2).

Key Concepts

Point Masses

In the realm of physics, the term point masses refers to objects that can be treated as if all of their mass is concentrated at a single point. This simplification is especially useful when calculating the center of mass for a system with multiple bodies. For example, in the given exercise, we are working with a collection of point masses, each with distinct mass values of 12 kg, 6 kg, 4.5 kg, and 15 kg, and specified coordinates representing their locations.

In real-world applications, this concept allows for the modeling of complex systems like planets orbiting a star or molecules in a gas as if they are simply points in space with mass, significantly simplifying calculations while providing sufficiently accurate results for many purposes.

Weighted Coordinates

The concept of weighted coordinates is related to the idea of center of mass. When we discuss weights in this context, we're not talking about barbells and dumbbells, but rather how each point mass in a system exerts influence depending on its mass and distance from a reference point.

Here’s the crunchy part: to calculate the weighted coordinates, each mass (\(m_i\)) is multiplied by its corresponding coordinate (\(x_i, y_i\) for x and y coordinates, respectively). The product of mass and coordinate gives us the 'weighted' coordinate. It's much like saying, 'The heavier the point mass, the more pull it has in determining the center of mass's position.' The solution to the exercise perfectly demonstrates this process as the first two steps, resulting in a new set of coordinates that has factored in the influence of each point mass.

Total Mass

To grasp the concept of total mass, think of it as the combined heft of all the point masses in the system. It is the summation of the mass of each point mass, regardless of its position in space. In the provided exercise, we add up each mass to arrive at the figure of 37.5 kg.

Why care about total mass? It plays a pivotal role when we calculate the center of mass. Imagine you're sharing a seesaw with friends; the total mass will decide how the seesaw balances, and in the same way, the total mass of a system influences the center of mass. It's a cumulative measure that's fundamental for subsequent calculations in determining the central point where one could balance the entire mass of the system.

Coordinate System

Let's dive into the coordinate system. This is your navigational star in the realms of mathematics and physics. It's like a map for locating points in a space, which can be two-dimensional like your classroom floor, or three-dimensional like the space in your room. In our textbook problem, we use a two-dimensional coordinate system where each point is defined by an ordered pair (\(x, y\) coordinates).

Moving forth, a fixed point called the origin, usually denoted (\(0,0\) ), serves as the starting block. From this point, we measure the 'address' of each mass in our problem using the given coordinates. As we did in our solution, it helps us provide a structured method to calculate the center of mass—essentially pinpointing where the tug-of-war between masses evens out and where our see-saw would balance perfectly if these masses were placed on it.