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7.4: Angular Momentum Conservation Model

Overview

The ideas we have developed for linear momentum and impulse apply to rotational motion as well. But first, we will need to develop the rotational analogs of the various variable and constructs we have been using. Force, momentum, velocity, impulse all have rotational analogs. The concept that impulse equals change in linear momentum has its analog in rotational motion as does the principle of conservation of momentum.

In the last model, we focused both on the properties of forces and the momentum transfers governing the connection of force to motion. We found that forces can be rather tricky to deal with, and we, hopefully, began to appreciate the usefulness of being very precise about technical terminology as it relates to force and motion and to the usefulness of representations such as momentum charts and force diagrams.

Now we extend the formalism to enable us to analyze and make sense of the motion of extended objects that can rotate as well as translate. We also introduce the last conserved quantity that we will work with, angular momentum (which could also be called rotational momentum). We will introduce a couple of additional concepts: torque and rotational inertia as well as to ways to describe rotational motion. We will then be in a position to answer detailed questions and make specific predictions about the magnitudes of individual forces and the changes in motion caused by the applied forces in a wide variety of situations.

Angular momentum is analogous to momentum (translational or linear momentum) even though they are quite different physical quantities. For instance, we found in the last model that the momentum of an object is conserved if there is no net external force acting on it. In this model we will find that the rotational analogue of force is called torque and that the angular momentum of an object is conserved if there is no net external torque acting on it (even if there is a net force). Similarly, a transfer of angular momentum is called angular impulse. Remember from Part 1 that work is the integral of the applied force over the distance the system moves. In this model we broaden our idea of work a little by including the energy transferred if a torque is applied over the angle that the system rotates.

However, translational or linear momentum (usually just called momentum) and angular momentum are clearly very different physical quantities and you will have to work hard and be careful at keeping them separate in your thinking. The difference is obvious when you see a physical situation but, when discussing abstract ideas without a physical picture in mind, it is easy to confuse the two quantities. For instance, a ball may be spinning (i.e. have angular momentum) and flying through the air in a straight line (i.e. have momentum). Or, it may be spinning at any speed (have any angular momentum) and not be flying through the air. Or, it may be flying through the air but not spinning at all. So, you see that the amount of angular momentum the ball has is completely independent of its momentum. The moral of this little story is the same as with all physics problems: try to keep a concrete physical picture in your head as you learn new abstract ideas.

The Center of Mass Idea

You may have realized by now that modeling objects as point particles is a rather drastic oversimplification, but often very useful. When does the extended geometry of a non-point object become important? Focusing on just one point of an object can describe perfectly adequately the translational motion of that object, but it does not tell us anything about the object’s rotation. Whether an object rotates or not, depends on where forces are applied to the object. We will not derive or prove the general result, described in the following paragraphs, that we use to handle this situation: combined translation and rotation of rigid objects. We will simply state it.

It turns out that we can consider all of the forces acting on the object as if they acted at one point, the center of mass, as far as translation is concerned. That is, if we are concerned only about an object’s translation, it doesn’t matter where the forces act on the object. We can consider them all to act at a single point! This is truly a great simplification. We have been using this result throughout this course without making a “big deal” about it. The special point where we consider the forces to act is called the center of mass. It is the same as the center of gravity (where you can support the object and it won’t rotate) as long as the gravitational force is uniform. Near the surface of the Earth, for all objects of ordinary size, the gravitational force can certainly be considered uniform, so for all problems we consider, the center of mass and center of gravity are the same point.

Now, what about rotations? To take into account the effect of applied forces on the rotation of an object, we have to know where the forces are applied. We use a new construct, the torque, \(\tau\) , which takes into account the magnitude and direction of the applied force as well as its distance from the point or axis about which the object rotates. If objects are constrained to rotate about a particular axis, such as a wheel mounted to an axle, the torques are typically computed about that axis. If there is no constraint, torques should be computed about the center of mass, the point about which the object will rotate.

In order to properly discuss the rotational analog of momentum, we need to develop a consistent way to describe rotational motion. We find an analogous set of rotational motion variables to translational motion variables. We will introduce these motional variables by looking at both the circular motion of a point object and the rotational motion of an extended object (an extended object has size, so it is not a point object). By dividing up the general motion of a rigid object into translation plus rotation, we can separately discuss the momentum (actually the translational momentum) and the angular momentum.

The Detailed Description of Rotational Motion—Rotational Kinematics

We begin by developing some useful relationships to describe the motion of a point object. Rather than using rectangular coordinates to describe the position P of a particle moving in a circle, we find it convenient to use polar coordinates, \(r\) and \(\Theta\) . The coordinate r is the distance of the point from an axis of rotation (the origin  \(\vartheta\) );  \(\Theta\) is the angular displacement from an arbitrarily chosen axis that defines zero. As in the figure 7.4.1, \(\Theta\) is frequently measured from the positive x axis, but it could be measured from any reference line.

Figure 7.4.1

When \(\Theta\) changes by an amount \(\Delta \Theta\) , the particle moves an amount \(\Delta s\) along the circumference of the circle defined by the radius \(r\). The arc length, \(\Delta s\) is simply the product of \(r\) and \(\Delta \Theta\) (Figure 7.4.2) :

\[\Delta s = r \Delta \Theta \tag{7.4.1} \]

Figure 7.4.2

The instantaneous velocity of the point P is always tangential to the curve at that point. If we differentiate the displacement with respect to time to get the tangential velocity of this object, we get an expression that depends only on the time derivative of \(\Theta\) :

\[\frac{ds}{dt} = v_{tangential} = r\frac{ d\Theta}{dt} . \tag{7.4.2} \]

The time rate of change of the angular position, \(\Theta\) , is called the \(angular~ velocity\) or \(rotational ~velocity\) and is usually represented by the Greek letter  \(\Omega\) \( ("omega")\).

\[\Omega = \frac{d\Theta}{ dt} .\tag{7.4.3}\]

The rotational velocity and tangential velocity are related by:

\[v_{tangential}= r\Omega \tag{7.4.4}\] .

The Units of \(\Theta\) and \(\Omega\) .

The units of \(\Theta\) and \(\Omega\) are respectively an angle unit and an angle unit divided by time. We can use any units we want and that are useful for a particular application for \(\Theta\) and \(\Omega\) . Typical units are degrees, degrees/second; revolutions, revolutions/second or rpm or revolutions/hour, etc. The "natural" units, are, however, radians and radians per second. We must use radians and radians per second when we use the relations connecting \(v\) to \( \Omega \), etc. Note that a “radian” is a rather “funny” kind of unit. For instance, radians multiplied by meters is just meters, not radian·meters. It is a useful word to put into sentences to tell us we are talking about angular motion (and to make phrases “sound right”), but it does not behave like a “real” unit such as meter or second.

Note that so far we have been discussing a point object constrained to move in a circle. We can also describe the kinematics of \(any\) extended object (e.g. a baseball bat) that is rotating about a fixed origin (where we grip it) by \(\Theta\) and \(\Omega\) , as long as we define the polar coordinates about the fixed axis of rotation. Actually, we can use this same approach for objects that are rotating as well as moving translationally, if we define the polar coordinates about the “center of mass.”

The Directions of \(\Theta\) and \(\Omega\) .

Just as the translational variables position, \(r\), and velocity, \(v\), have both direction and magnitude, so do the angular variables \(\Theta\) and \(\Omega\) . It is useful to treat these variables as vectors, \(\Theta\) and \(\Omega\) . What direction do these variables point? The only unique direction in space associated with a rotation is along the axis of rotation. So, if the axis of rotation gives the direction, we need only specify which way along the axis corresponds to a particular direction of rotation. By convention, the direction is specified by the “right-hand-rule.” If you curl the fingers of your right hand in the direction of positive \(\Theta\) or the direction rotation is occurring, your thumb points in the direction (along the axis of rotation) of \(\Theta\) or \(\Omega\) . We will see several more examples of the right-hand-rule (RHR).

When forces act on extended objects, they not only cause the object to change its translational motion, but can also cause it to change its rotational motion. That is, these forces can cause an angular acceleration as well as a translational acceleration. It turns out that it is not just the magnitude and direction of the force that is important in causing angular accelerations, but also where the force is applied on an extended object. Torque is the construct that incorporates both the vector force as well as where it is applied to an object.