# Kinetic energy of a body rotating on another rotating body

Consider a body which can freely rotate with respect to the inertial frame, and a rotating disk whose axis is fixed in body frame.

When applying the lagrangian method (does that make a difference?), is the kinetic energy of the disk with respect to the body frame (for constant disk rotation rate, the kinetic energy is constant) or with respect to inertial frame (for constant disk rotation rate, the kinetic energy depends on the rate of the parent body)?

Parent discussion: Defy gravity torques with gyroscopes?

## 1 Answer

It's obvious that the kinetic energy must be compute with respect to the inertial frame. Using Lagrangian method for your problem is very similar to applying this method in order to derive the equations of motion of a gyroscope that can be found anywhere (for example here).

The only difference between your case and a gyroscope is that in your case, the absolute spinning rate of the disk consists of two parts. The first is the spinning rate of disk with respect to body frame, and the the second is the projection of the angular velocity vector of body frame on disk axis, i.e. the absolute spinning rate of the disk is summation of these two parts.

Also the absolute precession and nutation rate of the disk are made by angular velocity of body frame.

• I had indeed identified 2 parts to the rate vector of the disk, I just wanted to confirm that when computing the kinetic energy the rate considered was wrt inertial frame (that is, combination of 2 rotations). I don't understand the second part though: isn't it supposed to be [rate of disk in body frame] + [rate of body in inertial frame]? Can you draw a diagram? – Mister Mystère Jul 6 '14 at 17:23
• @MisterMystère Yes, exactly. As you say the absolute angular velocity vector of disk is: [angular velocity vector of disk in body frame] + [angular velocity vector of body in inertial frame]. But what I mentioned was about the similarity between your case and a gyroscope. As you know we can separate the angular velocity of a gyroscope in three parts: spinning rate, nutation rate and precession rate. The spinning rate of disk wrt body frame does not create nutation or precession for disk. So the body frame rotation will produce these two rates. {To be continued...} – Mas ooD Jul 7 '14 at 11:28
• On the other hand, in situation that axis of disk and angular velocity vector of body frame are parallel, the absolute spinning rate of disk will be: [spinning rate of disk wrt body]+[magnitude of angular velocity of body frame] . So in general, the absolute spinning rate of disk is:[spinning rate of disk wrt body frame]+[projection of angular velocity vector of body frame on disk axis] . Inform me if my explanation isn't clear and I'll draw a diagram as soon as possible. – Mas ooD Jul 7 '14 at 11:28
• @MisterMystère, Why don't you use Euler equation ($\frac{d\vec{L}}{dt}=\vec{M}_{ext}$) instead of lagrange method. I think Euler equation will be much easier to handle for your problem. – Mas ooD Jul 7 '14 at 11:39