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Angular Momentum: The Concepts

Newton's Law's don't explicitly say anything about angular momentum. However, Newton's Laws may be extended to deal with rotational motion by defining the concept of Angular Momentum. We will also see that in analogy with the conservation of linear momentum (or just momentum, for short), there is also conservation of angular momentum under certain circumstances. Interesting physical effects are explained in terms of conservation of angular momentum.

Let's begin with some basic ideas and then examine some applications.

• The simplest definition of angular momentum follows from considering an object with mass m that moves in a circle with radius r and with a tangential velocity v. Its angular momentum L is given by L = rmv.
• Angular momentum is a vector, like Force, momentum and velocity. Its direction is determined by the so-called right hand rule. Picture the circle above and think of the velocity vector causing the object to rotate counter-clockwise. Allow the fingers of your right hand to follow the direction of the object. Your thumb pointing upward is the direction of the angular momentum.
• Angular momentum may also be defined in a form similar to linear momentum. While linear momentum is P = MV, where M is mass and V is velocity, angular momentum L = Iw, where I is rotational inertia and w (we use w instead of small Omega, the conventional symbol) is angular velocity. Angular velocity is just the angle the mass rotates in an interval of time. w has the units of radians/second.
• The tangential velocity (v) of the object going in a circle may be expressed in terms of w by the relationship v = rw. Note if w represents an angular velocity of 2pi radians/second, 2pi x r is just the circumference of a circle, so at that rate of rotation the object makes one complete circle in one second.
• The angular momentum of the object going in a circle may be expressed in terms of the angular velocity (w) by using v = rw in the definition L = rmv. Substituting for v, we get L = mr2w. But we also said L = Iw, where I is the rotational inertia. So for the case of the object of mass m at radius r, I = mr2. This shows how the rotational inertia depends not only on the mass, but where the mass is located relative to the center of rotation. For a more complex object, like a figure skater, I is not so simple, but it has the same general form, i.e., the mass of the object multiplied by some distance squared. So an object with a relatively small mass can have the same moment of inertia as an object with much greater mass if the small mass is distributed far from the center of rotation while the large mass object has its mass concentrated close to the center of rotation. (Think about how this explains the ice skater who spins rapidly by by bringing her arms close to her body.)
• Newton's Second Law for linear motion says the net force on the object
F = Delta P / Delta t = m Delta v / Delta t. The analogue for the object moving in a circle says the net torque acting on the object
T = Delta L / Delta t = I Delta w / Delta t. The torque is the twisting action caused by a force (F) tangent to the circle so that T is defined to be T = rF.
• Note the special case when Torque T = 0. This implies that Delta L is also zero. But Delta L = L(final) - L(initial) = 0, so L(final) = L(initial). This is conservation of angular momentum.

R.S. Panvini
9/30/2001