91Ó°ÊÓ

Point-masses are individual masses considered to be concentrated at a single point in a space. In many physics and engineering problems, complex objects are simplified as point-masses to make calculations more manageable. Think of point-masses like individual dots on a grid, each representing a certain amount of weight at a specific location. In our exercise, we have three point-masses located on an XY plane:

• Mass 1: 2 kg at the point (4, -1)
• Mass 2: 7 kg at the point (-2, 0)
• Mass 3: 5 kg at the point (-8, -5)

Each mass contributes both to the total mass of the system and to the moments about the axes. By having distinct positions, these point-masses influence the overall balance point or center of mass of the system. As you explore point-masses, imagine how each one's location and size affect where the entire system might balance on a single point.

Calculating the total mass of a system is one of the simplest but most essential steps to finding the center of mass. To determine the total mass, simply sum up the weights of all point-masses. This cumulative value helps us understand the entire system's mass without worrying about individual contributions afterward.

In our exercise, we have:

• Mass 1: 2 kg
• Mass 2: 7 kg
• Mass 3: 5 kg

By adding these masses together, we find the total mass:

• Total mass, \( m = 2 + 7 + 5 = 14 \) kg

This 14 kg represents the complete weight of the system, playing a crucial role in finding how this mass is distributed spatially through moments around axes.

The moment about the y-axis, denoted as \( M_{x} \), measures how far, and with what mass, each point-mass is from the y-axis. To find this, multiply each mass by its x-coordinate and sum the results. This

• Mass 1's moment: \( 2(4) = 8 \)
• Mass 2's moment: \( 7(-2) = -14 \)
• Mass 3's moment: \( 5(-8) = -40 \)

Add these up to get the total moment about the y-axis:

• \( M_{x} = 8 - 14 - 40 = -46 \)

A negative moment indicates that the combined mass acts more strongly on one side of the y-axis. This affects the center of mass's x-coordinate, informing us how off-center the mass is distributed around the y-axis.

To understand how mass is distributed relative to the x-axis, we calculate the moment about this axis, denoted as \( M_{y} \). This involves multiplying each mass by its y-coordinate, indicating how the mass contributes to balancing around the x-axis.

• Mass 1's moment: \( 2(-1) = -2 \)
• Mass 2's moment: \( 7(0) = 0 \)
• Mass 3's moment: \( 5(-5) = -25 \)

Summing these values gives us:

• \( M_{y} = -2 + 0 - 25 = -27 \)

Similar to the moment about the y-axis, the negative value here signifies a tendency for the system's center of mass to lean more in the negative direction along the y-axis. Understanding these moments allows us to pinpoint the center of mass, revealing the system's balance point in a clear and precise manner.