PHY 206 Chapter 10 Conservation of Angular Momentum / Dr. Miner

Two Second Laws - Most General Form

For translational motion we write the Second Law in the most general form as F = dP/dt. Here F and P are vectors, and t is a scalar. For rotational motion the second law is t = dL/dt. Similarly, t and L are vectors, and t is a scalar.

Two Second Laws - Special Cases

The above laws can be rewritten if one assumes constant mass for one, and constant rotational inertia for the other. That is, dm/dt = 0, and dI/dt = 0. Then F = dP/dt becomes m (dv/dt) + v (dm/dt) which is F = m (dv/dt) = m a. Here F, P and v are vectors; t is a scalar. Similarly, t = dL/dt becomes I (dw/dt) + w (dI/dt) which gives t = I (dw/dt) = I a. Here w is omega, the angular velocity. In the equation torque, L and w are vectors; t is a scalar.

The Vector Product or Cross Product

Consider two arbitrary three-dimensional vectors A and B. They may point in any directions. We will define the Vector Product, or Cross Product, as:

A × B = A B sin(q) u_{perpendicular}

In the above note that A, B and u_{perpendicular} are vectors; while A, B and sin(q) are scalars. u_{perpendicular} is a unit vector perpendicular to both vectors A and B. The angle q is measured from the first vector to the second. Thus if we reverse the vectors, noting that sin(-q) = - sin(q), we find:

A × B = - B × A

Unit Vectors

It is also possible to write the two vectors in i, j, k notation.

A = A_x i + A_y j + A_z k

B = B_x i + B_y j + B_z k

Note that the unit vectors i, j and k are mutually perpendicular. This simplifies the process since we note that the cross product of a unit vector with itself is equal to (1)(1)sin(0) = "0," whereas the cross product of two different unit vectors is always ±1. Note that sin(90^o) = "1." Thus i × i = j × j = k × k = 0; and i × j = j × k = k × i = - j × i = - k × j = - i × k = 1. The Cross Product of the two vectors is therefore:

A × B = (A_y B_z - A_z B_y) i + (A_z B_x - A_x B_z) j + (A_x B_y - A_y B_x) k

Thus the result of this multiplication of these two vectors is always a vector.

Angular Momentum as a Vector Product

We can define Angular Momentum, L, with L = r × P, where the "×" represents the vector cross product. If L, r, and P are mutually perpendicular vectors, then the equation can be rewritten as L = r P.

Torque as a Vector Product

In a move similar to the above, we define torque as a vector cross product: t = r × F, where again the "×" represents the vector cross product. Here t, r and F are vectors. If t, r and F are mutually perpendicular vectors, then the equation can be rewritten as t = r F, as in our earlier definition of torque.

Conservation of Angular Momentum

Noting that from t = dL/dt, if the net torque is zero on a system then its total angular momentum is constant, or conserved. Angular momentum can be written as L = I w, where w represents angular velocity. One interesting possibility is that L can remain constant, while both I and w change. For example the ice skater who pulls in her hands and thus reduces I, starts to spin faster, that is, has an increase in w.

Conservation of Energy Including Rotational Kinetic Energy

Previously we found that energy cannot be created or destroyed. It can, however, be changed from one type to another. We now have an additional type of energy, rotational kinetic energy (RKE). It is an energy like any other type and can participate in the Conservation of Energy process. Conservation of Energy includes all types of energy that we have seen (and will see) including but not limited to kinetic, rotational kinetic, elastic, gravitational and work (energy into or out of a system). Remember, "conservation" here means that the total of all types is constant, or conserved. It does not describe using energy sparingly to lessen the "energy crisis."

Quantization of Angular Momentum

In Atomic Physics we find that angular momentum is quantized, that is, the orbital angular momentum of electrons in atoms is always an integer multiple of 1.05 × 10^-34 joule-second. This number is Planck's Constant divided by 2p, and is often called "h-bar." This and other related matters will be important in describing the atom in the third semester of general physics.