6.4.9 Time derivative of the inertia tensor
When we analyze motion of a rigid body, we will need to calculate the time derivatives of the linear and angular momentum. Linear momentum is no problem, but for angular momentum, we will need to know how to differentiate IG with respect to time. There is a formula for this:
where W = dR/dt RT is the spin tensor (see sect 6.2.2)
6.4.9 Time derivative of angular momentum
To use the angular momentum conservation equation, we will need to know how to calculate the time derivative of the angular momentum. When we do this for a 3D problem, we need to take into account that the mass moment of inertia changes as the body rotates. We will prove the following formula:
dh/dt = rG × MaG +IGα + ω× IGω )
For 2D planar problems this can be simplified to:
dh/dt = rG × MaG + IGzzαz k
Proof: We start by taking the time derivative of the general definition of h
dh/dt = d/dt (rG x MvG +IGω )
We can go ahead and do the derivative with the product rule:
We can simplify this by noting that drG / dt =vG and of course the cross product of vG with itself is zero. We can also use the definition of angular acceleration: dω / dt = α . This gives
Finally we can use the formula for dIG / dt from 6.4.8 to see that
and recalling the spin tensor-angular velocity formula Wu = ω ×u for all vectors u.
where we recall that the cross product of a vector with itself is always zero. This gives the answer stated.