# BNC Scientific's T-Shirt Equation Explained

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.

## Want one? “Like” us on Facebook.The first 50 people will be entered to win a T-shirt.

## 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 *w _{s}*) to the vectors

*H*and

*w*. A simple way to do this is with the vector dot product:

_{s}Where:

*H* has been replaced with *H _{G}* (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

|*H _{G}*| is the magnitude of the vector

*H*

_{G}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:

Where:

*i*, *k* are unit vectors pointing along the positive *x* and *z* axis, respectively

*w _{x}*,

*w*,

_{y}*w*are the components of the angular velocity vector of the object (with respect to ground), resolved along the

_{z}*x*,

*y*,

*z*directions, respectively

*I _{x}*,

*I*,

_{y}*I*are the principal moments of inertia about the

_{z}*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 *I _{x}* =

*I*,

_{z}*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

*H*, and as a result

_{G}*w*=

_{x}*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 *I _{x}* =

*I*≡

_{z}*I*, let's now find an equation relating

_{w}*w*and

_{s}*w*. Since

_{p}*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 *w _{p}* and

*w*we get

_{s}Hence, *w _{p}* and

*w*are constant.

_{s}If we eliminate *H _{G}* 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.**