.ipynb .pdf

repository open issue

Binder JupyterHub

Contents

import numpy as np
import matplotlib.pyplot as plt

from IPython.lib.display import YouTubeVideo
YouTubeVideo('FPg5EjHEo6c')

Building Quiz_06

A $2-m\times 2-m \times 2-m$ cube is attached at a corner to a spherical joint. It can rotate along the x-, y-, and z-axes, but the attached corner cannot move. The moment of inertia about its center of mass is

$$\begin{split}I_G = \frac{m}{12} \left[\begin{array} ~8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{array}\right]~m^2\end{split}$$

a. Use the parallel axis theorem to shift the point of rotation to corner where the cube is fixed.

b. What is the kinetic energy if the cube has an angular velocity of $\mathbf{\omega}=1\hat{i}+10\hat{j}$

I_G = 8/12*np.eye(3,3)
I_G

array([[0.66666667, 0.        , 0.        ],
       [0.        , 0.66666667, 0.        ],
       [0.        , 0.        , 0.66666667]])

Parallel axis theorem

$$\begin{split}I_O = m \left[\begin{array} ~y^2+z^2 & -xy & -xz \\ -xy & x^2+z^2 & -yz \\ -xz & -yz & x^2+y^2 \end{array}\right]~m^2 + I_G\end{split}$$

x = -1
y = -1
z = -1
I_parallel = np.array([[y**2+z**2, -x*y, -x*z],
                       [-x*y, x**2+z**2, -y*z],
                       [-x*z, -y*z, x**2+y**2]])
I_parallel

array([[ 2, -1, -1],
       [-1,  2, -1],
       [-1, -1,  2]])

IO = I_parallel + I_G
IO

array([[ 2.66666667, -1.        , -1.        ],
       [-1.        ,  2.66666667, -1.        ],
       [-1.        , -1.        ,  2.66666667]])

Kinetic Energy

Kinetic energy because the point of rotation is fixed,

$T_O = \frac{1}{2}\mathbf{\omega I_O \omega}$

omega = np.array([1, 10, 0])

T = 0.5*omega@IO@omega

T

124.66666666666664

hO = IO@omega
hO

array([ -7.33333333,  25.66666667, -11.        ])

previous

Building Quiz_05

next

Building Quiz 07