Problem 10 - APEX Calculus

Question

Point masses are given along a line or in the plane. Find the center of \(\operatorname{mass} \bar{x}\) or \((\bar{x}, \bar{y}),\) as appropriate. (All masses are in grams and distances are in cm.) $$ \begin{array}{l} m_{1}=1 \text { at }(-1,-1) ; \quad m_{2}=2 \text { at }(-1,1) ; \\ m_{3}=2 \text { at }(1,1) ; \quad m_{4}=1 \text { at }(1,-1) \end{array} $$

Step-by-Step Solution

Answer

Center of mass is at (0, 1/3).

Step 1: List the given masses and their coordinates

We are given four point masses with the following values and coordinates:\- Mass \( m_1 = 1 \) at \((-1, -1)\)\- Mass \( m_2 = 2 \) at \((-1, 1)\)\- Mass \( m_3 = 2 \) at \((1, 1)\)\- Mass \( m_4 = 1 \) at \((1, -1)\)

Step 2: Calculate the total mass

The total mass is the sum of all individual masses:\[ M = m_1 + m_2 + m_3 + m_4 = 1 + 2 + 2 + 1 = 6 \text{ grams} \]

Step 3: Compute the weighted sum of the x-coordinates

The formula for the weighted sum of x-coordinates is:\[ \text{Sum of } mx = m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4 \]Substitute the given values:\[ \text{Sum of } mx = (1)(-1) + (2)(-1) + (2)(1) + (1)(1) = -1 - 2 + 2 + 1 = 0 \]

Step 4: Compute the weighted sum of the y-coordinates

The formula for the weighted sum of y-coordinates is:\[ \text{Sum of } my = m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4 \]Substitute the given values:\[ \text{Sum of } my = (1)(-1) + (2)(1) + (2)(1) + (1)(-1) = -1 + 2 + 2 - 1 = 2 \]

Step 5: Calculate the x-coordinate of the center of mass

The x-coordinate of the center of mass is given by:\[ \bar{x} = \frac{ \text{Sum of } mx }{ M } = \frac{0}{6} = 0 \]

Step 6: Calculate the y-coordinate of the center of mass

The y-coordinate of the center of mass is given by:\[ \bar{y} = \frac{ \text{Sum of } my }{ M } = \frac{2}{6} = \frac{1}{3} \approx 0.333 \]

Step 7: Finalize the center of mass

The center of mass for the system of point masses is located at coordinates:\[ (\bar{x}, \bar{y}) = (0, \frac{1}{3}) \]

Key Concepts

Weighted Sum

To find the center of mass, one essential concept we use is the *weighted sum*. This is a way to calculate the overall effect of different masses positioned at various points. For example, think of it like this: If you have different weights placed on a see-saw, the heavier side would tip down and the lighter side up. The weighted sum helps us balance these masses. For coordinates of mass locations, we calculate it by multiplying each mass by its respective coordinate value: - For the x-coordinate, multiply each mass by its x-position. - Similarly, for the y-coordinate, multiply each mass by its y-position. This calculation helps us understand where the majority of the weight lies in relation to the axis. Once all these individual products are added together, we obtain the *weighted sum* for either the x or y coordinates.

Total Mass

The *total mass* is relatively straightforward but fundamental. It represents the sum of all individual masses in a system. Consider it like collecting coins. Each coin has its value, and when all are gathered together, they show the complete worth, just as the total mass shows the entire weight.In our problem, the total mass is:- Add up each of the masses provided: \[ M = m_1 + m_2 + m_3 + m_4 \]- In this instance, summing them up gives us 6 grams.Understanding the total mass is crucial as it provides a denominator to our center of mass calculations, balancing the equations and ensuring they are correctly scaled.

x-coordinate of the Center of Mass

The *x-coordinate of the center of mass* pinpoints the exact horizontal location on an imaginary balance line, where the entire mass could be placed and still remain in balance. Imagine this place as the exact middle part of a tightly stretched string holding up different weights.To calculate it, use this formula:- Divide the weighted sum of the x-coordinates by the total mass: \[ \bar{x} = \frac{\text{Sum of } mx}{M} \]- Using our problem, this calculation results in: \[ \bar{x} = \frac{0}{6} = 0 \]Having a zero result means that horizontally, the center of mass lies right at the center between the positions given. It is like finding a center point on a line of beads strung evenly across a wire.

y-coordinate of the Center of Mass

Similar to the x-coordinate, the *y-coordinate of the center of mass* tells us the precise vertical position on an equilibrium scale, offering a complete picture of balance within a plane. Imagine adjusting the height level to ensure no side dips below the other.To find the y-coordinate, the formula used is:- Take the weighted sum of the y-coordinates and divide it by the total mass: \[ \bar{y} = \frac{\text{Sum of } my}{M} \]- For our given masses and locations, it simplifies to: \[ \bar{y} = \frac{2}{6} = \frac{1}{3} \approx 0.333 \]This indicates that vertically, the center of mass is slightly elevated above the coordinate origin, showing a predominance of mass in the upward direction. It's like a see-saw with a small weight slightly favoring one side, creating a gentle tilt.