A rigid body of spheroidal shape, spinning rapidly about its axis of symmetry, is placed on a smooth flat table. Show by considering the moment of the force at the point of contact that its axis will precess in one direction if it is oblate \((ca=b, e . g\). a rugby ball). Show also that if there is a small frictional force, the axis will become more nearly vertical, so that if the body is oblate its centre of mass will fall, but if it is prolate it will rise.

Gyroscopic torque is the pivotal concept behind the mesmerizing precession movements of spinning bodies, like our spheroidal example. When a rigid body rotates, it creates angular momentum. If a force, like gravity, acts on this spinning body at a point away from its axis of rotation, this interaction results in a torque known as gyroscopic torque. Now, imagine spinning a top; the gyroscopic torque is why it doesn’t just fall over as gravity is pulling it down, but instead, it precesses around a point on the surface.

Gyroscopic torque is always directed perpendicular to the plane formed by the gravitational force and the angular momentum vector. This perpendicular direction dictates the precession direction, which is essential for understanding the behavior of oblate and prolate spheroids on a surface. In the case of a spheroid, the mass distribution because of its shape will influence the direction of precession, introducing the fascinating difference in behavior between objects like a discus and a rugby ball.

The moment of inertia is fundamentally a measure of an object’s resistance to changes in its rotation. It depends on both the mass of the object and its distribution relative to the axis of rotation. To grasp this concept, think of a figure skater spinning with arms outstretched; the skater will spin slower compared to when arms are pulled in close to the body—the distribution of mass changes, and so does the moment of inertia. Similarly, an oblate spheroid (like a discus) has its mass distributed farther from the axis of symmetry, leading to a larger moment of inertia, whereas a prolate spheroid (like a rugby ball) has a smaller moment of inertia due to a closer mass distribution to the axis.

The moment of inertia is crucial because it not only influences the speed at which an object rotates but also interacts with external forces like gravity to create gyroscopic torque, an interplay that eventually impacts the precession of our spinning spheroids.

Angular momentum is a vector quantity that represents the rotational equivalent of linear momentum. It depends on the rotation speed – the angular velocity (\f\(\rhoomega\f\)) and the moment of inertia (\f\(I\f\)). The angular momentum of a spinning object can be expressed as \f\(L = I\fwomega\f\). For an object that is free to rotate in space (like our spheroid on a smooth table), the conservation of angular momentum states that its angular momentum remains constant unless acted upon by an external torque.

In the case of the spinning spheroid, the angular momentum is determined by its rotation speed and shape, and it plays a critical role in the way the spheroid precesses. The gyroscopic torque that arises as a result of the gravitational force and the angular momentum vector influences the precession pattern and stability of the spheroid. Understanding the relationship between the moment of inertia, angular velocity, and angular momentum is pivotal in predicting the motion of rigid bodies.

Frictional forces might seem trivial but can have profound effects on the dynamics of spinning objects. Even a small frictional force at the point of contact of a spinning spheroid with a table can alter the motion significantly. Friction opposes relative motion and in doing so, generates a torque that can change the orientation of the spinning axis.

For oblate and prolate spheroids, frictional forces cause an additional torque that results in the axis of rotation tilting towards becoming more vertical. In oblate spheroids, this torque causes a reduction in the height of the center of mass because the axis tilts in such a way that it 'falls' towards the horizontal orientation. Conversely, for prolate spheroids, this torque 'lifts' the center of mass as the axis tilts towards the vertical. It's a subtle interaction of forces, but understanding the effect of friction on the precession, stability, and orientation of a spinning body is critical for interpreting the behavior of such systems in real-world applications.