# 4.2. Classical Mechanics¶

## 4.2.1. Rigid Body Rotation¶

In all the sections below, the rigid body is rotating around the axis, so: ### Kinetic Energy¶

The kinetic energy is: where is the total angular momentum: ### Angular Momentum¶

Total angular momentum is: Where is the moment of inertia tensor: ### Moment of Inertia¶

The moment of inertia tensor and its components are: Let’s write (where is a unit vector), then the kinetic energy is: where is the moment of inertia about the axis of rotation: #### Cylinder¶

Solid cylinder of radius , height and mass . We’ll use cylindrical coordinates. First for rotation about the axis: Code:

>>> from sympy import var, integrate, pi
>>> var("m V R rho z phi h")
(m, V, R, rho, z, phi, h)
>>> I = m/V * integrate(rho**2 * rho, (rho, 0, R), (phi, 0, 2*pi), (z, -h/2, h/2))
>>> I.subs(V, pi * R**2 * h)
R**2*m/2


And about the axis: Code:

>>> from sympy import var, integrate, pi, cos
>>> var("m V R rho z phi h")
(m, V, R, rho, z, phi, h)
>>> I = m/V * integrate((rho**2+z**2-rho**2*cos(phi)**2) * rho, (rho, 0, R), (phi, 0, 2*pi), (z, -h/2, h/2))
>>> I.subs(V, pi * R**2 * h).simplify()
m*(3*R**2 + h**2)/12


Special cases are a rod of length (set above) and a thin solid disk of radius and mass (set above).

#### Sphere¶

Solid sphere of radius and mass . We’ll use spherical coordinates. All axes are equivalent, so we use rotation about the axis: Code:

>>> from sympy import var, integrate, pi, sin
>>> var("m V R rho theta phi")
(m, V, R, rho, theta, phi)
>>> I = m/V * integrate(rho**4 * sin(theta)**3, (rho, 0, R), (phi, 0, 2*pi), (theta, 0, pi))
>>> I
8*pi*R**5*m/(15*V)
>>> I.subs(V, 4*pi*R**3/3)
2*R**2*m/5