2. A Little Physics of Rotational Motion

In order to understand why the gyro starts preceding instead of falling it is necessary to grasp the notions of torque and angular momentum, and the equation relating them.

2.1 Torque

The effect of applying two forces of the same magnitude, but acting on different lines, to a rigid body, is to make the latter rotate. As this effect is unique and, although related to the concept of force, quite different from it, physicists have given it its own name and definition. A torque (or moment) of a force about a reference point (O, in Fig.3) is defined as the cross product between the position vector of the point of application of the force and the force vector :

So the line on which t lies is along the normal to the plane formed by r and F, and points in a direction given by the right hand rule (or right handed screw rule). Its magnitude is t=rFsinq, where theta is the angle between r and F.

2.2 Angular Momentum

When a body of mass m is moving with velocity v, it is extremely useful to express its linear momentum, defined as p=mv. It's difficult for me to explain the meaning of linear momentum, it relates the inertia of a body, with its movement. But I need not discuss this further here. What needs to be mentioned is the conservation of linear momentum. Newton's second law states, in its "momentum" form, that the force applied to a body equals the rate of change of momentum of the body: F=dp/dt. It can be easily shown that if F is the net force on a system of particles, it will produce an overall change of momentum dp on the system in a time dt. And, it's clear, that if F=0, then p is going to be constant: the momentum of the system is conserved(!). Angular momentum relates with the angular velocity of a positon vector pointing at the body from a reference point (or line, or plane). It is defined in a way which is analougous to the torque. But, as we shall see, its physical behaviour is much more similar to that of linear momentum (it too can be conserved). The angular momentum of a body with a momentum p about a point O is given by: L = r x p Where everything is like in the above defintion of the torque, only the force is replaced by the linear momentum.

2.3 The equation of rotational motion

But some may ask: if we can deal with changes of linear momentum as an exernal force, can we not do the same with momentum???. The answer is (yes!:) letÕs try and differentate L and see what we get:

dL/dt = dr/dt x p + r x dp/dt : the first term dr/dt x p(is just)= v x mv = 0 (vv sin0!); also we konw that F=dp/dtso we have: dL/dt = r x F, but the latter is just the torque about O: r x F = t so that, finally:

In words: when a torque wrt (with respect to) a point O is applied to a body the rate of change of angular momentum of the body (moving in relation to a point O) is equal to such a torque. This result is beautifully simple, and (naturally) analogous to the relation between force and linear momentum. As for Newton II, it is also valid for a system of particles (which includes our gyro!), if t is the torque of the external forces acting on the system, causing a change of angular momentum dL, in a time dt. The next natural step (which gets us closer to gyroscopes) is to infer that if t = 0 then angular momentum is conserved (just like linear momentum when F=0!).

3. Rigid bodies

Index