Torque Free Motion

Torque-free motion refers to the movement of a rigid body without any external torques acting upon it.

Consider an axisymmetric object, with the z-axis being the axis of symmetry, having the center of mass defined at the origin of the rotating frame of reference. Here, the moment of inertia about the x and y axes are equal.

Consider the inertial frame of reference defined such that the positive Z-axis of the inertial frame is along the angular momentum vector and makes an angle θ with the positive z-axis of the rotating frame.

So, the angular momentum can be expressed in terms of unit vectors in two ways, and by equating the components of unit vectors, the equation for the object's angular velocity is derived.

Additionally, writing the angular velocity in terms of angular displacement and equating the components again gives the equation of motion for a torque-free axisymmetric rigid body.

Here, the object's angular momentum, precession, and spin remain constant throughout the motion along with angle θ.