Problem 8 - 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.) $$ \begin{array}{l} m_{1}=2 \text { at } x=-3 ; \quad m_{2}=2 \text { at } x=-1; \\ m_{3}=3 \text { at } x=0 ; \quad m_{4}=3 \text { at } x=7 \end{array} $$

Step-by-Step Solution

Answer

The center of mass \(\bar{x}\) is 1.3 cm.

Step 1: Identify the Masses and Positions

List the masses and their respective positions along the line. We have four point masses:- Mass \(m_1 = 2\) grams at position \(x_1 = -3\)- Mass \(m_2 = 2\) grams at position \(x_2 = -1\)- Mass \(m_3 = 3\) grams at position \(x_3 = 0\)- Mass \(m_4 = 3\) grams at position \(x_4 = 7\)

Step 2: Calculate the Total Mass

The total mass \(M\) is the sum of all individual masses:\[M = m_1 + m_2 + m_3 + m_4 = 2 + 2 + 3 + 3 = 10 \text{ grams}\]

Step 3: Compute the Moment About the Origin

The moment \(M_x\) about the origin is calculated by multiplying each mass by its position and then summing all the products:\[M_x = m_1 \cdot x_1 + m_2 \cdot x_2 + m_3 \cdot x_3 + m_4 \cdot x_4 = 2 \cdot (-3) + 2 \cdot (-1) + 3 \cdot 0 + 3 \cdot 7\]\[M_x = -6 - 2 + 0 + 21 = 13\]

Step 4: Find the Center of Mass \(\bar{x}\)

Divide the total moment by the total mass to find the center of mass:\[\bar{x} = \frac{M_x}{M} = \frac{13}{10} = 1.3\, \text{cm}\]

Key Concepts

Point Mass

When dealing with the center of mass, it's important to understand the concept of a point mass. A point mass is a simplification where an object's entire mass is assumed to be concentrated at a single point. This makes complex calculations easier, especially in physics and engineering.

• Each point mass is represented by its mass value, such as 2 grams or 3 grams in our exercise.
• The position of each point mass is given along a line, like at coordinates -3, -1, 0, and 7 in this example.
• This model helps in calculating the center of mass for systems where objects have specific locations and mass distributions.

Understanding the idea of point masses is crucial for solving problems involving multiple objects because it allows one to calculate moments and centers of mass with relative simplicity.

Moment about Origin

The moment about the origin is a critical part of finding the center of mass in a system of point masses. The moment is essentially a measure of how much each mass contributes to the system's balance based on its position relative to the origin, which is often set at zero on the coordinate plane.

• To calculate the moment, multiply each mass by its respective position (the distance from the origin).
• Sum all these products to get the total moment, noted as \(M_x\).
• For instance, in this exercise, you compute the moment as: \[M_x = 2 \times (-3) + 2 \times (-1) + 3 \times 0 + 3 \times 7 = 13\]

Moments give us insights into how different masses and their placements affect the overall balance and center of mass of a system.

Total Mass

In calculating the center of mass, determining the total mass of the system is essential. Total mass is simply the sum of all the individual masses in the system. This exhibits how the overall weight of an assembly affects the position of its center of mass.

• Add up each of the point masses to find the total mass \(M\).
• For the problem here: \(M = 2 + 2 + 3 + 3 = 10\) grams.
• This value helps in calculating the average position of the mass distribution.

By understanding the total mass, you combine the mass from each point into a single value, which is then used in further computations to find the distribution's balance point or center of mass.