Question

Five equal point masses are placed at the five corners of a square pyramid whose square base is centered on the origin in the \(x y\) plane, with side \(L,\) and whose apex is on the \(z\) axis at a height \(H\) above the origin. Find the CM of the five-mass system.

Short answer

The center of mass is at \((0, 0, \frac{H}{5})\).

Step-by-step solution

Identify Positions of Masses

The pyramid's base is a square located in the \(xy\)-plane and centered at the origin. The corners of the square are \((\frac{L}{2}, \frac{L}{2}, 0)\), \((-\frac{L}{2}, \frac{L}{2}, 0)\), \((-\frac{L}{2}, -\frac{L}{2}, 0)\), and \((\frac{L}{2}, -\frac{L}{2}, 0)\). The apex is above the origin at \((0, 0, H)\).

Calculate Center of Mass for X-Axis

The \(x\)-coordinates for the masses are \(\frac{L}{2}, -\frac{L}{2}, -\frac{L}{2}, \frac{L}{2},\) and \(0\). The total mass is \(5m\). The center of mass in the x-direction is calculated as:\[\bar{x} = \frac{m(\frac{L}{2}) + m(-\frac{L}{2}) + m(-\frac{L}{2}) + m(\frac{L}{2}) + m(0)}{5m} = 0\]

Calculate Center of Mass for Y-Axis

The \(y\)-coordinates for the masses are \(\frac{L}{2}, \frac{L}{2}, -\frac{L}{2}, -\frac{L}{2}, 0\). The center of mass in the y-direction is:\[\bar{y} = \frac{m(\frac{L}{2}) + m(\frac{L}{2}) + m(-\frac{L}{2}) + m(-\frac{L}{2}) + m(0)}{5m} = 0\]

Calculate Center of Mass for Z-Axis

The \(z\)-coordinates for the masses are \(0, 0, 0, 0, H\). The center of mass in the z-direction is:\[\bar{z} = \frac{m(0) + m(0) + m(0) + m(0) + m(H)}{5m} = \frac{H}{5}\]

Compile Results for Center of Mass

The overall center of mass for the system is:\[(\bar{x}, \bar{y}, \bar{z}) = (0, 0, \frac{H}{5})\].

Key concepts

Coordinate Geometry

Coordinate geometry helps us map different geometrical figures and understand their positions in space using a coordinate system. In this exercise, we deal with a square pyramid based on the Cartesian coordinate system.
The coordinates are important for pinpointing where each mass lies. The square base is on the xy-plane, represented by the coordinates

• \((\frac{L}{2}, \frac{L}{2}, 0)\)
• \((-\frac{L}{2}, \frac{L}{2}, 0)\)
• \((-\frac{L}{2}, -\frac{L}{2}, 0)\)
• \((\frac{L}{2}, -\frac{L}{2}, 0)\)

All four points are equidistant from the center, which is the origin (0, 0, 0).
The apex, which is above the base, lies on the z-axis at \(0, 0, H\). Placing these points in the coordinate system allows us to understand and visualize how the mass is distributed across the structure.

Mass Distribution

Understanding how mass is distributed is crucial in analyzing the center of mass. In our pyramid, all five point masses are equal, which simplifies calculations.
The masses at the four corners of the base contribute equally in the xy-plane, which is why the center of mass in both the x-direction and y-direction is zero. The apex mass lies above the origin, affecting the center of mass in the z-direction.
In essence,

• The equal mass at each vertex simplifies the symmetry.
• All points on the base contribute to a balanced mass distribution around the origin.
• The apex’s mass distinguishes the system’s center of mass more in the vertical (z) direction.

This understanding underscores how different points in a structure affect the overall position of the center of mass, especially when symmetry and equal mass distribution are involved.

Pyramid Structure

A pyramid is a fascinating geometrical shape characterized by its polygonal base and triangular faces that converge at a single point, the apex. In this particular problem, the base is a square where equal point masses are placed.
The apex is vertically above the center of the base, which introduces a unique point compared to the planar distribution of the base. The structure:

• Has a central apex above the origin making it three-dimensional.
• Forms a balance between the base and apex, contributing to how the center of mass is calculated.
• Helps visualize the balance of mass in relation to height, showing how it shifts based on the structure not being flat but rather elevated.

Understanding the structure is essential in predicting how mass and height affect the center of mass in three-dimensional figures.

Symmetry in Physics

Symmetry plays a pivotal role in physics, simplifying complex problems using uniformity and predictability principles. In our pyramid problem, symmetry is quite evident, especially given the equal distribution of masses at symmetric points.
This symmetry:

• Means that masses on the base counterbalance each other's x and y coordinates, resulting in a center of mass at the origin for these dimensions.
• The symmetry helps simplify calculations because the predictable distribution negates any influence in lateral directions, thanks to canceling effects.
• The symmetry in mass distribution aids in focusing on the z-axis for the center of mass shift.

By understanding symmetry, not only does the calculation become straightforward, but it also allows us to predict the system behavior, even before doing the math.