# Spinning Chair and Wheel

1. Nov 19, 2014

### hydroxide0

1. The problem statement, all variables and given/known data
The problem is Problem 2 on page 3 here: http://ocw.mit.edu/courses/physics/...ngular-momentum/MIT8_01SC_problems25_soln.pdf

2. Relevant equations
3. The attempt at a solution

My question concerns the solution in the above link. According to the solution, the angular momentum of the wheel about the central axis is
$${\bf L}_{S,w}^{\text{total}}={\bf L}_{S,w}^{\text{rot}}+{\bf L}_{cm,w}^{\text{spin}}=((I_w+m_wd^2)\omega+I_w\omega_s){\bf \hat{k}}.$$
(See page 4.) However, isn't the angular momentum of a system of particles about a point equal to the angular momentum of the center of mass plus the angular momentum of the particles with respect to the center of mass? (See http://en.wikipedia.org/wiki/Angular_momentum#Angular_momentum_simplified_using_the_center_of_mass, for example.) This would mean the angular momentum of the wheel about the central axis is actually
$${\bf L}_{S,w}^{\text{total}}=(m_wd^2\omega+I_w\omega_s){\bf \hat{k}}.$$

Thanks to anyone who can help clear this up!

2. Nov 20, 2014

### voko

There are multiple problems with the proposed solution. One is a fairly common mistake that the angular velocity of a rigid body depends on the reference frame, as in "the angular velocity of the person and the stool about a vertical axis passing through the center of the stool" and "the wheel is also spinning about its center of mass with angular velocity". That is wrong; a rigid body's angular velocity is invariant; see (31.3) in https://archive.org/stream/Mechanics_541/LandauLifshitz-Mechanics#page/n105/mode/2up and http://en.wikipedia.org/wiki/Angular_velocity#Consistency.

Another is the assumption that the "spin angular momentum", whatever that means, remains unchanged in magnitude; this is, at best, unjustified.