Problem 258 - Calculus Volume 1

Question

For the following exercises, calculate the center of mass for the collection of masses given. $$ m_{1}=1 \text { at }(1,0) \text { and } m_{2}=4 \text { at }(0,1) $$

Step-by-Step Solution

Answer

The center of mass is at \(\left(\frac{1}{5}, \frac{4}{5}\right)\).

Step 1: Identify the Formula

The formula to find the center of mass for point masses in two dimensions is given by: \[(x_{cm}, y_{cm}) = \left( \frac{\sum m_i x_i}{\sum m_i}, \frac{\sum m_i y_i}{\sum m_i} \right)\]where \(m_i\) are the masses and \((x_i, y_i)\) are their coordinates.

Step 2: List Given Values

We have two masses \(m_1 = 1\) located at \((1, 0)\) and \(m_2 = 4\) located at \((0, 1)\).

Step 3: Calculate Total Mass

The total mass \(M\) is the sum of the individual masses: \[M = m_1 + m_2 = 1 + 4 = 5\].

Step 4: Calculate the X-Coordinate of Center of Mass

Substitute the known values into the formula for the x-coordinate of the center of mass: \[x_{cm} = \frac{m_1 x_1 + m_2 x_2}{M} = \frac{1 \times 1 + 4 \times 0}{5} = \frac{1}{5}\].

Step 5: Calculate the Y-Coordinate of Center of Mass

Substitute the known values into the formula for the y-coordinate of the center of mass: \[y_{cm} = \frac{m_1 y_1 + m_2 y_2}{M} = \frac{1 \times 0 + 4 \times 1}{5} = \frac{4}{5}\].

Step 6: Write the Coordinates of the Center of Mass

The center of mass for the given masses is at \(\left(\frac{1}{5}, \frac{4}{5}\right)\).

Key Concepts

Calculation of Center of Mass

Calculating the center of mass helps you determine the point where a collection of masses can be balanced perfectly. The formula you use for this purpose in two dimensions is \[(x_{cm}, y_{cm}) = \left( \frac{\sum m_i x_i}{\sum m_i}, \frac{\sum m_i y_i}{\sum m_i} \right)\]. This means, you'll need to perform an average for both the x and y coordinates of the masses, factoring in their weights. This approach allows the position of each mass to influence the location of the center of mass based on its size
.

• First, find each coordinate's contribution by multiplying it with the respective mass.
• Then, sum these contributions for all masses in your system.
• Finally, divide by the total mass of the system to find the center coordinates.

By understanding this calculation, you'll grasp how differently positioned and weighted points contribute to a system's equilibrium.

Point Masses in Two Dimensions

Point masses are simply masses that are concentrated at single points in space. In our exercise, they sit at specified coordinates in a two-dimensional plane. This setup allows you to treat each object as having no size, simplifying how you account for their position when calculating the center of mass
.

• The position of each mass is defined by coordinates \(x_i, y_i\).
• Each mass seems equivalent to a pinpoint with all its weight localized there.
• Using coordinates can visually and numerically analyze their placements.

This method makes it easy to apply algebraic formulas and generate precise results in geometric distributions across a flat space.

Mathematics Problem-Solving

Solving problems in mathematics becomes easier once you understand the logic and reasoning behind the operations. The calculation of the center of mass is a great example of mathematical problem-solving. It involves:

• Understanding the formula: What does each component do?
• Analyzing the given values: Identifying the positions and weights of each mass.
• Applying arithmetic: Performing calculations to derive the solution.

Remember, mathematical problem-solving benefits greatly from practice and repetition. Each step builds upon the previous one, so double-checking your work helps to catch errors early and reinforce understanding.

Coordinate Geometry

Coordinate geometry is a vital part of modern mathematics. It enables you to plot points, lines, and curves on a plane using coordinates. This framework is crucial for understanding the problem at hand, where the goal is to determine the center of mass by placing our point masses on a coordinate plane. Think of it as a blueprint that relates algebraic equations to geometric figures
.

• Coordinate systems allow you to work with points defined by pairs of numbers, like \(x_i, y_i\) for locations of masses.
• Geometric interpretation helps visualize how points interact and influence the center of mass.
• Using graph paper or a plot can aid comprehension by providing a visual reference.

Having a strong grasp of coordinate geometry will make it easier to handle complex spatial problems such as calculating centers of mass for multiple points.