Problem 10 - Calculus of a Single Variable

Question

Find the center of mass of the given system of point masses. \(\begin{array}{|c|c|c|c|}\hline m_{i} & {10} & {2} & {5} \\\ \hline\left(x_{i}, y_{i}\right) & {(1,-1)} & {(5,5)} & {(-4,0)} \\\ \hline\end{array}\)

Step-by-Step Solution

Answer

The center of mass of the given system is located at \((-2/17, 0)\)

Step 1: Calculation of X component of the center of mass

To calculate the X component of the center of mass \(\bar{X}\), use the formula \(\bar{X} = \frac{\sum m_{i} x_{i}}{\sum m_{i}}\). Plugging the given values, this yields \(\bar{X} = \frac{(10*1) + (2*5) + (5*(-4))}{10 + 2 + 5}\).

Step 2: Calculation of X component of the center of mass

To compute the Y component of the center of mass \(\bar{Y}\), use the formula \(\bar{Y} = \frac{\sum m_{i} y_{i}}{\sum m_{i}}\). Plugging the given values, this yields \(\bar{Y} = \frac{(10*(-1)) + (2*5) + (5*0)}{10 + 2 + 5}\).

Step 3: Simplification of \(\bar{X}\) and \(\bar{Y}\)

Simplify both expressions to get the center of mass. For \(\bar{X}\), we obtain \(\bar{X} = \frac{-2}{17}\) and for \(\bar{Y}\), we get \(\bar{Y} = \frac{0}{17}\).

Key Concepts

Point Masses

Point masses are idealized objects where all the mass of the object is assumed to be concentrated at a single point in space. This simplification makes it easier to calculate properties like the center of mass.
Such calculations are especially useful in systems where the mass is distributed in discrete particles rather than being continuously spread out over an object.

• In our problem, each mass, denoted as \( m_i \), is located at a specific coordinate \( (x_i, y_i) \).
• These masses are 10 at \((1,-1)\), 2 at \((5,5)\), and 5 at \((-4,0)\).

By treating these as point masses, calculating the center of mass becomes straightforward, as we don't have to worry about the shape or size of the individual masses.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows us to use algebra to solve geometric problems.
In our context, the coordinates \( (x_i, y_i) \) are crucial to locating each point mass within the plane.

• Using these coordinates, we can calculate geometric properties like the center of mass.
• The formula for the center of mass takes into account the coordinates of each point and its mass, weighing each position by its mass.

This approach not only provides an accurate location of the center of mass but also demonstrates how mass influences the geometric center of a system in space.

Mass Distribution

Mass distribution describes how mass is spread over a system. It influences the calculations of key physical properties such as the center of mass.
In our point mass system, mass distribution is given by the individual masses and their respective positions.

• The mass-weighted average of the positions gives us the center of mass, which reflects the point where the system can be balanced.
• For example, more massive points will have a larger impact on the center of mass than less massive ones, effectively pulling the center of mass closer to their location.

Understanding mass distribution helps explain how changing the mass or position of any point in the system will shift the center of mass.