Chapter 11 - ANGULAR MOMENTUM

A.     Conservation  of Angular Momentum

                                        

The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero, that is, if the system is isolated.

This follows directly from Equation  which indicates that if

Than       

Ltot = constant or Li = Lf

For an isolated system consisting of a number of particles, we write this conservation law as L_tot = ΣL_n = constant, where the index n denotes the nth particle in the system. If the mass of an isolated rotating system undergoes redistribution in some way, the system’s moment of inertia changes. Because the magnitude of the angular momentum of the system is L = Iω, conservation of angular momentum requires that the product of I and ω must remain constant. Thus, a change in I for an isolated system requires a change in ω. In this case, we can express the principle of conservation of angular momentum as

I_iω_i = I_fω_f. = constant

Therefore, the angular momentum about the center of mass must be conserved—that is, I_iω_i = I_fω_f. For example, when divers wish to double their angular speed, they must reduce their moment of inertia to half its initial value

we have a third conservation law to add to our list. We can now state that the energy, linear momentum, and angular momentum of an isolated system all remain constant:

B.     The Motion of Gyroscopes and Tops

A very unusual and fascinating type of motion you probably have observed is that of a top spinning about its axis of symmetry, as shown in Figure 11.14a. If the top spins very rapidly, the symmetry axis rotates about the z axis, sweeping out a cone (see Fig. 11.14b). The motion of the symmetry axis about the vertical—known as processional motion—is usually slow relative to the spinning motion of the top.

The essential features of precessional motion can be illustrated by considering the simple gyroscope shown in Figure 11.15a. The two forces acting on the top are the downward gravitational force Mg and the normal force n acting upward at the pivot point O. The normal force produces no torque about the pivot because its moment arm through that point is zero. However, the gravitational force produces a torque τ =  r x Mg about O, where the direction of τ is perpendicular to the plane formed by r and Mg. By necessity, the vector τ lies in a horizontal xy plane perpendicular to the angular momentum vector. The net torque and angular momentum of the gyroscope are related through.

 

To simplify the description of the system, we must make an assumption: The total angular momentum of the precessing wheel is the sum of the angular momentum Iω due to the spinning and the angular momentum due to the motion of the center of mass about the pivot. In our treatment, we shall neglect the contribution from the center-of-mass motion and take the total angular momentum to be just Iω. In practice, this is a good approximation if ω is made very large.

The vector diagram in Figure 11.15b shows that in the time interval dt, he angular momentum vector rotates through an angle d , which is also the angle through which the axle rotates. From the vector triangle formed by the vectors    and   , we see that

where we have used the fact that, for small values of any angle θ, sin  θ ≈ θ. Dividing through by dt and using the relationship L = Iω, we find that the rate at which the axle rotates about the vertical axis is

The angular speed ω_p is called the precessional frequency. This result is valid only when  ω_p « ω. Otherwise, a much more complicated motion is involved. As you can see from Equation 11.20, the condition ω_p « ω is met when ω is large, that is, the wheel spins rapidly. Furthermore, note that the precessional frequency decreases as ω increases—that is, as the wheel spins faster about its axis of symmetry.

Suppose that the spacecraft carries a gyroscope that is not rotating, as in Figure 11.16a. In this case, the angular momentum of the spacecraft about its center of mass is zero. Suppose the gyroscope is set into rotation, giving the gyroscope a nonzero angular momentum. There is no external torque on the isolated system (spacecraft + gyroscope), so the angular momentum of this system must remain zero according to the principle of conservation of angular momentum.

C.     Angular Momentum as a Fundamental Quantity

We have seen that the concept of angular momentum is very useful for describing the motion of macroscopic systems. However, the concept also is valid on a submicroscopic scale and has been used extensively in the development of modern. theories of atomic, molecular, and nuclear physics. In these developments, it has been found that the angular momentum of a system is a fundamental quantity. The word fundamental in this context implies that angular momentum is an intrinsic property of atoms, molecules, and their constituents, a property that is a part of their very nature.

To explain the results of a variety of experiments on atomic and molecular systems, we rely on the fact that the angular momentum has discrete values. These discrete values are   multiples of the fundamental unit of angular momentum where h is called Planck’s constant:

            Fundamental unit of angular momentum =

Equating the angular momentum to the fundamental unit Ћ , we can find the order of magnitude of the lowest angular :

  we found that the moment of inertia of the O2 molecul e about this

axis of rotation is 1,95 x 10^-46 kg.m². Therefore :

The Danish physicist Niels Bohr (1885–1962) accepted and adopted this radical idea of discrete angular momentum values in developing his theory of the hydrogen atom. Strictly classical models were unsuccessful in describing many of the hydrogen atom’s properties. Bohr postulated that the electron could occupy only those circular orbits about the proton for which the orbital angular momentum was equal to nh, where n is an integer. That is, he made the bold claim that orbital angular momentum is quantized. One can use this simple model to estimate the rotational frequencies of the electron in the various orbits.