Caucasian female ice skater in a blue one-sleeved ice skating outfit with skirt, doing a jump on the ice.

Spinning ice skaters change their moment of inertia to change the speed of their spin. | Image Credit: iStock.com/Artur Didyk

Conservation of Momentum in Rotational Motion

Module 8 introduced the importance of using the moment of inertia for a rigid object when studying rotation. You learned about the rotational equations for kinematics and kinetic energy, and you learned that rotational dynamics uses torque, which is the analog of force.

Just as momentum is conserved in straight line collisions, momentum of rotation is conserved. This is called angular momentum, and the laws of physics dictate that angular momentum is a conserved quantity:

L with rightwards arrow on top subscript i n i t i a l end subscript space equals space L with rightwards arrow on top subscript f i n a l end subscript

If you’ve ever watched an ice skater perform a spin, you’ve no doubt noticed how she can increase or decrease her rate of rotation at will. How can she change her rate of rotation if she’s not pushing on the ice to make herself go faster? If you pay close attention while the skater is spinning, you’ll notice that she is changing the shape of her body by bringing her arms closer to her body or extending them. In other words, she’s changing her moment of inertia.

Recall that moment of inertia is equivalent to mass, so the skater’s ability to change her moment of inertia is like an object in motion changing its mass. In linear motion momentum, rho with italic rightwards arrow on top italic space italic equals italic space m v with italic rightwards arrow on top is conserved if there are no outside forces acting on it. What would happen to an object with constant linear momentum if that object lost mass? Because momentum is conserved, if mass decreased, then velocity would increase. That’s exactly what happens with the skater. Her moment of inertia decreases so her angular velocity increases. Because is equivalent to , and is equivalent to , it makes sense that the angular momentum of a rigid object must be I omega . The symbol straight L is used for angular momentum, so you can write

L italic space italic equals italic space I omega

and the law of conservation of angular momentum can be written

L with rightwards arrow on top subscript i n i t i a l end subscript space equals space L with rightwards arrow on top subscript f i n a l end subscript

This very simple analysis will go a long way toward helping you understand and solve many types of problems involving angular motion. However, there are some important features to angular momentum (and from Module 8, torque) that you need to revisit to understand these problems more completely.

Screwdriver screwing a screw into wood

Turning a screwdriver is an example of applying torque. | Image Credit: iStock.com/Jason_V

Torque and Angular Momentum as Vectors

Torque and angular momentum are the rotational analogs of force and linear momentum. Force and momentum are vectors, so does that mean that torque and angular momentum are also vectors? Yes—but to understand how to make sense of a rotational vector, we need to learn a new bit of math called the vector product.

Recall that the dot product produces a scalar quantity from two vectors. The vector product, on the other hand, produces a vector from two input vectors. This third vector has the special property that it is perpendicular to both input vectors, and its magnitude is the product of the magnitudes of the input vectors times the sine of the angle between them. Perhaps the mention of sine sounds familiar? Recall that

tau space equals space F r space sin space phi

This quantity is the magnitude of the torque. To express torque as a vector, we use the vector product, sometimes called the cross product:

tau with rightwards arrow on top space equals space stack r space with rightwards arrow on top space cross times F with rightwards arrow on top

So, the torque vector is perpendicular to both the force vector and the moment arm vector. The vector product has many special properties that are useful for calculations. (Refer to your textbook to review some of them.)

Further, the angular momentum of a particle as a vector (see Module 8) can be written as

L with italic rightwards arrow on top italic space italic equals italic space stack r italic space with italic rightwards arrow on top italic cross times p with italic rightwards arrow on top

And finally, just as force is the rate of change of linear momentum, p with rightwards arrow on top , torque, is the rate of change of angular momentum:

$tau with rightwards arrow on top space equals space fraction numerator d L with rightwards arrow on top over denominator d t end fraction$

If angular momentum and torque are to be considered vectors, then what is the direction of the vector? This is a subtle question because, by definition, the entire system is rotating and therefore constantly changing direction. Recall that the vector product yields a vector that’s perpendicular to the two input vectors—but in which direction, up or down? Physicists have decided that, for consistency, the result of a vector product will be a vector in the direction that follows the right-hand rule. Extend your right hand and let your index finger point in the direction of the first vector of the vector product and let your middle finger point in the direction of the second vector. With those two fingers properly aligned your extended thumb points in the direction of the third vector, the vector product. Your textbook has helpful illustrations of the right-hand rule for vector products.

Watch the following video to view an example of using the vector product to determine torque.

Cross Product and Torque. Khan Academy, 2011. YouTube. Web. Accessed 20 April 2018

Rotating gyroscope on a white background

A spinning gyroscope stays upright because of angular momentum. | Image Credit: iStock.com/DNY59

Gyroscopes

You may not be an ice skater, but conservation of angular momentum affects your everyday life. You might have even benefited from this conservation law today. Have you ever ridden a bicycle? Have you ever thought about why a bicycle is so stable? How is it possible to ride a bike, even very slowly, and stay upright? It’s all about angular momentum. The angular momentum vector of the spinning bicycle wheel is directed along the wheel’s axle, and it does not want to change direction. Just as any change in momentum requires a force—or, in the case of angular momentum, a torque—in the absence of that outside influence, the angular momentum vector stays pointing in the same direction and you get to stay on your bicycle.

This same principle applies to a gyroscope, often encountered in the form of a child’s toy or science lab demonstration tool. The gyroscope is a wheel with a massive rim and light spokes. It therefore has a large moment of inertia. Once the gyroscope starts spinning (from a torque applied by the user pulling a string wrapped around the axle), it has a large angular momentum and that L with rightwards arrow on top doesn’t want to change direction.

Of course, eventually the gyroscope (or equivalently, a child’s top) will slow down its spin due to friction. Then an odd thing happens to the gyroscope: It starts to rotate around a vertical axis while it continues to spin around its own axis of symmetry—a type of movement called precessional motion. Why is that?

To understand this, you really need to appreciate the vector nature of torque and angular momentum. Consider a top. While it’s spinning and standing perfectly upright, its angular momentum vector is pointing straight up and it doesn’t want to change. Let r with rightwards arrow on top be the displacement from the pivot point of the top (its tip) to the center of mass of the top. For this analysis, we consider the top as a point particle at . Because tau with rightwards arrow on top space equals space stack r space with rightwards arrow on top space cross times F with rightwards arrow on top , torque is zero when the top is straight up because F, the force of gravity, is down and r is either up or down depending whether the rotation is clockwise or counterclockwise. If two vectors are parallel or antiparallel, then their vector product is zero.

As the top slows, it becomes less stable and begins to fall over. Now, though, r and F are no longer parallel so their cross product and, therefore, the torque acting on the top, is no longer zero. Given that F is down, and r is in a vertically oriented plane, the cross product between the two is horizontal. The torque on that tipping top is horizontal and, therefore, parallel to the table it is spinning on. Because its tip is fixed by friction to not move on the table, the torque causes the top to slowly rotate in a circle, sweeping out a cone whose tip coincided with the tip of the top.

Watch the following video to explore what causes precession and other orbital changes.

What Causes Precession and Other Orbital Changes? Khan Academy, 2011. YouTube. Web. Accessed 20 April 2018.

Summary

You may want to refer back to this lecture for support on the following concepts:

Conservation of angular momentum
Torque and angular momentum as vectors
Precessional motion