Problem 1 - Calculus of a Single Variable

Question

Find the center of mass of the point masses lying on the \(x\) -axis. \(m_{1}=7, m_{2}=3, m_{3}=5\) \(x_{1}=-5, x_{2}=0, x_{3}=3\)

Step-by-Step Solution

Answer

The center of mass of the point-masses situated on the x-axis is at -1.

Step 1: Identify given values

The first thing to do is to distinguish all the known parameters. The three masses are \(m_{1}=7kg\), \(m_{2}=3kg\), and \(m_{3}=5kg\). The corresponding positions along the x-axis are \(x_{1}=-5\), \(x_{2}=0\), and \(x_{3}=3\).

Step 2: Compute total mass

Before finding the center of mass, calculate the total mass of the system by adding up all the individual masses: \(M = m_{1} + m_{2} + m_{3} = 7 + 3 + 5 = 15 kg\).

Step 3: Find the center of mass

Substitute the identified masses and positions into the formula for the center of mass and calculate: \(\bar{x} = \frac{1}{M} \sum m_i x_i = \frac{1}{15} (7*(-5) + 3*0 + 5*3) = -1\).

Key Concepts

Point Masses

Point masses are fundamental concepts in physics and engineering. A point mass is a perfectly small object with mass but no significant size or shape to affect its behavior in calculations. When we refer to point masses, we only care about two main aspects: the amount of mass and its position in space. This makes it easier to perform calculations compared to using objects with complex shapes.

• In this exercise, we have three point masses: 7 kg, 3 kg, and 5 kg.
• Each mass is located at a specific position on the x-axis: -5, 0, and 3 respectively.

Point masses help simplify the process of determining the center of mass by reducing complex shapes to simpler models. Even if objects are not actual point masses, they can still be treated as point masses for theoretical calculations.

X-axis

The x-axis is a crucial component in the Cartesian coordinate system, which is used to determine the position of points in space. It is a horizontal line where the position of objects or point masses is measured using a linear scale. In this exercise, all of our point masses lie on the x-axis, meaning we only need to consider their positions along this line.

• The x-axis helps to place and identify the exact positions of point masses with respect to the origin (0,0).
• Here, positions are given as -5, 0, and 3, describing how far and in which direction each mass is from the point 0.

Understanding the placement of point masses on the x-axis is vital for calculating the center of mass, as their positions directly influence the result.

Total Mass

Total mass is the sum of all individual masses in a system. It's an important parameter for finding the center of mass, as it helps to weigh the influence of each mass on the overall system.

• To compute the total mass, simply add up the masses of all point objects involved.
• In this exercise, the total mass is calculated as 15 kg, which is the sum of 7 kg, 3 kg, and 5 kg.

The concept of total mass provides a foundation for balance and stability in systems. A larger mass implies more influence on the system's behavior due to its gravitational pull.

Center of Mass Formula

The center of mass is the point at which the entire mass of a system can be considered to be concentrated. The formula to find the center of mass along a particular axis, like the x-axis, is essential in mechanics. The center of mass formula is given by:\[\bar{x} = \frac{1}{M} \sum m_i x_i\]Where:

• \(m_i\) are the individual masses.
• \(x_i\) are their corresponding positions.
• \(M\) is the total mass of the system.

In this specific problem, substituting the values into the formula gives us the center of mass at \(-1\). This value locates the point along the x-axis where the masses are evenly distributed. Understanding and applying the center of mass formula is crucial for determining balance points in various scientific and engineering contexts.