In physics, there are two different types of precession: torque-induced and torque-free. Torque induced precession is when an object spins equally from its axis when torque is applied (i.e. a spinning toy top). On the other side of physics, torque-free precession occurs when the axis of rotation is slightly different from the stable rotation of an axis. For example, when a Frisbee is thrown, the disc has a rotation around its axis, but its rotation does not remain symmetric to its axis of rotation. Another example of this would be our Earth, as it is subject to torque induced by the Sun and Moon’s pull on the Earth’s axis.
BNC Scientific approached our T-shirt design with torque in mind. Torque relates to our Earth, our moon, our Solar System. We live and breathe torque free motion, one might say. And yet, torque free motion equations aren’t easily recognized. And thus, our T-shirt design was born.
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NOW, IT’S YOUR TURN. CHECK OUT THE EQUATION BELOW AND LET US KNOW HOW YOU USE TORQUE DURING YOUR DAY.
Gyroscope Physics — Torque Free Motion
Consider the figure below with local xyz axes as shown.
Let's find an equation that relates the angle θ (between H and ws) to the vectors H and ws. A simple way to do this is with the vector dot product:
H has been replaced with HG (since we are using the angular momentum about the center of mass G of the object)
j is a unit vector pointing along the positive y axis
|HG| is the magnitude of the vector HG
Differentiate the above equation with respect to time to give
which gives the following equation for dθ/dt:
From vector derivative we know that:
From angular momentum:
i, k are unit vectors pointing along the positive x and z axis, respectively
wx, wy, wz are the components of the angular velocity vector of the object (with respect to ground), resolved along the x, y, z directions, respectively
Ix, Iy, Iz are the principal moments of inertia about the x, y, z directions, respectively
Substitute the above three equations into the equation for dθ/dt and we get
This is an informative equation coming out of the gyroscope physics analysis done here. It tells us that for Ix = Iz, dθ/dt = 0 (as the object rotates through space). But if we choose α = 0 then the precession axis coincides with the angular momentum vector HG, and as a result wx = dθ/dt = 0 (which simplifies the calculations). Hence, the angle θ is constant and this is why, from the point of view of an observer in the inertial reference frame, the precession axis appears to coincide with the angular momentum vector. But mathematically speaking it does not matter what axis we choose as the precession axis, since it is simply a component of rotation. Being able to arbitrarily choose the precession axis is similar to how you can arbitrarily choose the x,y directions for a force calculation. Ultimately the answer is the same and the resultant force is not going to change. To understand this better you can read up on Euler angles which are commonly used to define the angular orientation of a body, using the concept of precession, spin, and nutation (which have been used in the gyroscope physics analysis presented here).
Using the above result for Ix = Iz ≡ Iw, let's now find an equation relating ws and wp. Since θ is always constant we can express the angular momentum as follows in terms of its x,y,z components:
Now, from before
which can be written as (to match notation used previously):
We can equate the i,j,k components to give:
But from geometry we can also write:
Solving the above equations for wp and ws we get
Hence, wp and ws are constant.
If we eliminate HG from the above two equations we get
Note that this is the same as the equation given previously for uniform gyroscopic motion with negligible rod mass:
for the case L = 0. For L = 0, this equation reduces to torque free motion for an axisymmetric body.