Question

In Exercises 29 and \(30,\) find the center of mass of the given system of point masses. $$ \begin{array}{|l|c|c|c|} \hline m_{i} & 5 & 1 & 3 \\ \hline\left(x_{1}, y_{1}\right) & (2,2) & (-3,1) & (1,-4) \\ \hline \end{array} $$

Short answer

The center of mass of the system of point masses is at the coordinates (4/9, -5/9).

Step-by-step solution

Identify the Masses and Their Coordinates

From the table, it can be established that there are three point masses in the system: a mass of 5 at coordinates (2,2), a mass of 1 at coordinates (-3,1), and a mass of 3 at coordinates (1,-4).

Apply the Formulas for the X and Y Coordinates of the Center of Mass

Apply the formulas for the X and Y coordinates. To get X, multiply each mass with its x-coordinate and add them together, and then divide by the total mass: \( X = \frac{(5*2) + (1*-3) + (3*1)}{5 + 1 + 3} = \frac{4}{9} \). To get Y, do the same with the y-coordinates: \( Y = \frac{(5*2) + (1*1) + (3*-4)}{5 + 1 + 3} = -\frac{5}{9} \).

Verify and State the Final Result

The center of mass for this system of point masses is at the coordinates (4/9, -5/9). Always good to check each step for accuracy while executing the calculation.

Key concepts

Point Masses

Understanding the concept of point masses is fundamental when calculating the center of mass for any number of objects. In physics, a point mass is an idealization where an object is considered to have mass concentrated at a single point in space, disregarding its size and shape. This simplification is often used in physics to make complex problems manageable.

When dealing with a system of point masses, like in the given exercise, we treat each object as if all its mass were 'pointed' at the specific coordinates provided. Therefore, in the example provided, the masses of 5, 1, and 3 are each treated as being located at the points (2,2), (-3,1), and (1,-4), respectively. This greatly simplifies the process of finding the center of mass as we only need to concern ourselves with these specific points and the mass values associated with them.

Understanding that these are not distributed masses, but concentrated at points in space, helps to make the use of formulas for the center of mass straightforward and applicable.

Coordinate System

The coordinate system is a method for representing points in a specific space using numbers, which are known as coordinates. For the context of our problem, we use a two-dimensional Cartesian coordinate system where each point can be specified by two coordinates: the horizontal (x) position and the vertical (y) position.

In the exercise, the coordinates of the point masses are given in the Cartesian system as pairs (x, y). This allows us to precisely locate where each mass is in relation to the origin (0,0), which is usually where the two reference axes cross each other. The x-axis runs horizontally while the y-axis runs vertically, and the coordinates tell us how far along each of these axes the point mass lies.

For calculating the center of mass, using the coordinate system is invaluable. We need to take into account the positions (x and y coordinates) of all the masses, which we do by using weighted averages, where the weights are the masses of the point objects. With a clear understanding of coordinate systems, interpreting and solving such mathematical problems becomes much easier.

Mass Distribution

Mass distribution refers to how mass is spread out in space. For point masses, the mass distribution simplifies to discrete points with associated masses. However, in more complex objects, the distribution of mass could be continuous, and the shape of the object would significantly affect the calculation of the center of mass.

In the case of our textbook problem, the mass distribution is straightforward: each mass is concentrated at a single point and has no extent. To find the center of mass for a system like this, we consider the 'weighted' positions of each point mass, meaning each point's contribution to the center of mass is proportional to its mass. The larger the mass, the more it will 'pull' the center of mass towards its position.

It is also important to understand that the overall center of mass for the system represents the average position of the entire mass distribution. It is the balancing point where, if the entire mass of the system were concentrated, the distribution would be in equilibrium. By mastering these basic principles of point masses, the coordinate system, and mass distribution, students can effectively approach problems involving the center of mass.