Joseph Bell

University Physics I - Project Report

Introduction:

The center of mass is a position defined relative to an object or system of objects. If the particle masses are all the same, the center of mass should be at the center of the object. It is the average position of all the parts of the system, weighted according to their masses. An unconstrained object on which there is no net force rotates on the center of mass. For simple rigid objects with uniform density, the center of mass is located at the centroid. Sometimes the center of mass doesn't fall anywhere on the object. The center of mass of a ring for example is located at its center, where there isn't any material. For more complicated shapes, we need a more general mathematical definition of the center of mass which is the unique position at which the weighted position vectors of all the parts of a system sum up to zero. Center of mass is useful because it makes it easy to solve mechanics problems.

Discussion:

For our project we had to find the center of mass of various shapes and then of a system. Center of mass is a “point” in an object that all other objects rotate about. This point varies by object. The center of mass is an essential part of understanding rotational motion since once found it is used to denote an object as if all the mass of the object is concentrated on this point. Center of mass is dependent on two factors. The individual masses of the particles in the object and the individual particles distance from the zero point, or origin of the object. These individual particles are summed up and then divided by the total mass of all the particles. If the object exists in more than one dimension then the distance can be found using vectors and then broken down into its vector components. Due to the many particles that exist in an object we use sigma notation for discrete objects and for non discrete cases we use integration to find the center of mass.

Discrete:

$$x_{cm} = \frac{1}{M} \sum_{i}^{n}m_{i}x_{i} = \frac{m_{1}x_{1}+m_{2}x_{2}+...+m_{n}x_{n}}{m_{1}+m_{2}+...+m_{n}}$$

$$y_{cm} = \frac{1}{M} \sum_{i}^{n}m_{i}y_{i} = \frac{m_{1}y_{1}+m_{2}y_{2}+...+m_{n}y_{n}}{m_{1}+m_{2}+...+m_{n}}$$

$$z_{cm} = \frac{1}{M} \sum_{i}^{n}m_{i}z_{i} = \frac{m_{1}z_{1}+m_{2}z_{2}+...+m_{n}z_{n}}{m_{1}+m_{2}+...+m_{n}}$$

Non discrete:

For non discrete cases it is essential to understand that dm must be equated to a proportion such that this equivalent expression has dx, dy ,dz. The interval [a,b] needs to represent the full length, area, or volume on the object depending on how many dimensions it's in. This depends on the amount of dimensions of the object. For example, for a 3D shape the equivalent expression would be:

$dm = \frac{dV}{V}dx$

$x_{cm} = \frac{1}{M}\int_a^b x dm$

$y_{cm} = \frac{1}{M}\int_a^b y dm$

$z_{cm} = \frac{1}{M}\int_a^b z dm$

For our experiment we had the coordinates of discrete shapes with equal mass. This allowed us to replace M, the total mass, with n, the number of particles in the object. This means are equation is now $r = \frac{1}{n}\sum_{i}^{n}r$ this like the other discrete equations, can be broken down to its vectorial components x, y, z. After this we simply calculate the Center of Mass via the discrete equations. Since all the masses are equivalent then the Center of Mass for each shape will be the average coordinate of x, y, z components which would be the geometrical center of the object. For a cluster of particles we used the same concept. Even though each cluster is composed of many different particles by calculating the Center of Mass of each object and using the coordinates of each object’s center of mass. Afterwards we can treat each object’s motion by their Center of Mass. Therefore calculate the Center of Mass of the entire cluster by using the C.M of each object and our discrete formulas for Center of Mass.

Objective: For the system of clusters we were asked to find the center of mass of a system of clusters. Each cluster contained many particles of identical mass. The theory behind our code and our logic was based on the center of mass equations. Because center of mass allows one to treat any object, shape, as if the mass is concentrated at its center of mass. We used this logic to understand that if we could treat each individual cluster by its center of mass as opposed to using all of its points. Afterwards we were able to use the coordinates of the center of masses of each cluster and find the center of mass of the entire system. The other way to achieve this would have been to take the center of mass of all the individual particles of every cluster at once. The reason these two methods work is because you can equate an object to its center of mass.

def Center_Of_Mass(filename):
    """This fuction takes as input a file with the coordinates of the points
    and returns the coordinates of the center of mass of the distribution.
    
    PARAMETERS:
    ===========
    filename = Name of the file that has the coordinates of the points.
    
    RETURNS:
    ========
    A tuple with the coordinates of the center of mass and the coordinates of
    all the points in the distribution.
    
    """
    # Three arrays that hold values for X,Y, and Z coordinates of given particles.
    xval = []
    yval = []
    zval = []
    # Opens particle distribution file
    f = open(filename, "r")
    # Calculates center of mass for each column of given particle distribution and places it in according arrays
    for line in f:
        numbers = line.split()
        x = numbers[0]
        y = numbers[1]
        z = numbers[2]
        xval.append(float(x))
        yval.append(float(y))
        zval.append(float(z))
        sumX = sum(xval)
        sumY = sum(yval)
        sumZ = sum(zval)
        totalMassX = len(xval)
        totalMassY = len(yval)
        totalMassZ = len(zval)
        comX = sumX / totalMassX
        comY = sumY / totalMassY
        comZ = sumZ / totalMassZ
        #returns center of mass and every value in specified array
    return (comX, comY, comZ, xval, yval, zval)

# Creates 3D plot
from mpl_toolkits import mplot3d

import numpy as np
import matplotlib.pyplot as plt

%matplotlib notebook
%matplotlib notebook
# Uses function to find center of mass and plots that center of mass with red dot.
fig = plt.figure()
ax = plt.axes(projection="3d")
coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution.txt')
ax.scatter3D(coX, coY, coZ, c='red', cmap='hsv');
# Plots center of mass of each particle given and adds it to an array.
coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_0.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox = []
ycoy = []
zcoz = []
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_1.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_2.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_3.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_4.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_5.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_6.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_7.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_8.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)

coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_9.txt')
ax.scatter3D(coX, coY, coZ, c='blue', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='gray', cmap='hsv',alpha = 0.05);
xcox.append(coX)
ycoy.append(coY)
zcoz.append(coZ)


# Calculates center of mass of accumulated center of masses of each particle given.
comsumX = sum(xcox)
comsumY = sum(ycoy)
comsumZ = sum(zcoz)
comtotalMassX = len(xcox)
comtotalMassY = len(ycoy)
comtotalMassZ = len(zcoz)
com_of_x = comsumX / comtotalMassX
com_of_y = comsumY / comtotalMassY
com_of_z = comsumZ / comtotalMassZ
#Plots that center of mass of center of masses with green dot
ax.scatter3D(com_of_x, com_of_y, com_of_z, c='green', cmap='hsv');

Objective: The objective was to find the center of mass of a spherical distribution. This was done by break each vector to its vector components;x,y,z. Afterwards we calculated the center of mass coordinate for each vector component. Due to each mass being identical the total mass can be represent by the total number of particles in the spherical distribution. By doing this we were able to arrive at the center of mass.

def Center_Of_Mass(filename):
    """This fuction takes as input a file with the coordinates of the points
    and returns the coordinates of the center of mass of the distribution.
    
    PARAMETERS:
    ===========
    filename = Name of the file that has the coordinates of the points.
    
    RETURNS:
    ========
    A tuple with the coordinates of the center of mass and the coordinates of
    all the points in the distribution.
    
    """
    xval = []
    yval = []
    zval = []
    f = open(filename, "r")
    for line in f:
        numbers = line.split()
        x = numbers[0]
        y = numbers[1]
        z = numbers[2]
        xval.append(float(x))
        yval.append(float(y))
        zval.append(float(z))
        sumX = sum(xval)
        sumY = sum(yval)
        sumZ = sum(zval)
        totalMassX = len(xval)
        totalMassY = len(yval)
        totalMassZ = len(zval)
        comX = sumX / totalMassX
        comY = sumY / totalMassY
        comZ = sumZ / totalMassZ
        
    return (comX, comY, comZ, xval, yval, zval)

from mpl_toolkits import mplot3d

import numpy as np
import matplotlib.pyplot as plt

%matplotlib notebook
%matplotlib notebook
fig = plt.figure()
ax = plt.axes(projection="3d")
coX, coY, coZ, x_points, y_points, z_points,  = Center_Of_Mass('spherical_distribution_1.txt')
ax.scatter3D(coX, coY, coZ, c='red', cmap='hsv');
ax.scatter3D(x_points, y_points, z_points, c='purple', cmap='hsv',alpha = 0.05);