#!/usr/bin/env python
# coding: utf-8

# # Zeroth and First Moments of Mass Example
# 
# Find the distance $\mu$ such that the mass center is closest to the line $\hat{\mathbf{n}}$.

# In[1]:


import sympy as sm
import sympy.physics.mechanics as me
me.init_vprinting()


# In[2]:


m, u = sm.symbols('m, mu')


# In[3]:


A = me.ReferenceFrame('A')


# Locations of each particle:

# In[4]:


p1 = A.y + A.z
p2 = A.x + A.z
p3 = A.x + A.y


# Mass center vector:

# In[5]:


p_star = (m*p1 + 2*m*p2 + u*p3)/(m + 2*m + u)
p_star


# Define the line:

# In[6]:


n = (A.x + A.y + A.z)/sm.sqrt(3)
n


# Form a vector that has a magnitude equal to the distance of interest:

# In[7]:


D = p_star - (p_star.dot(n)*n)
D.simplify()


# In[8]:


D_mag = D.magnitude().simplify()
D_mag


# Find $\mu$ such that 

# In[9]:


D_mag.diff(u)


# In[10]:


sm.solve(D_mag.diff(u), u)[0]