Advanced Laboratory and/or Lecture Demonstration Apparatus
Apparatus Title: -- Why Cats Land on their Feet, a Demonstration of Conservation of Angular Momentum
Abstract: A falling cat's legendary ability to land on its feet depends on angular momentum conservation. Our "cat" is a cylinder with a motorized tail. When dropped, the tail spins, rotating the body in the opposite direction. Students are challenged to calculate the height to drop the cat for it to land on its feet.
Idealized Cat Model Demonstrates Angular Momentum Conservation
The legendary ability of cats to land on their feet when falling provides an interesting example of an application of the conservation of angular momentum. When a cat falls through a large enough distance, it is able to rotate to an upright orientation and then stop the rotation so it lands on its feet, even if it is dropped with its back down and its feet in the air and with no initial rotation. (Of course, landing on its feet is not necessarily a guarantee that the cat will be unharmed, but it does provide the greatest chance of survival.) The flexibility of a cat's skeleton and its tail provide the keys to this feat. A cat's front and rear quarters are able to move independently through a considerable angle of rotation and it is able to swing its tail while outstretched to the side (perpendicular to its body) in a full circle. This flexibility allows the cat to rotate parts of its body in one direction while other parts rotate in the opposite direction to maintain zero net angular momentum. This ability also, of course, depends a great deal on instinct in addition to flexibility.
When choosing examples to illustrate physical phenomena, it is most effective to use examples that build on some prior knowledge that the student has. The ability of cats to land on their feet is common knowledge to most students, even if the reasons are not. So a demonstration of this phenomenon with an accompanying explanation or lesson provides a connection to a real life situation that allows the students to internalize the information more completely.
In order to illustrate the principle of conservation of angular momentum, as it relates to the cat, we have created a very idealized working model of a cat. Our "cat" is intended to be used as an interactive workshop or laboratory demonstration where the students, working collaboratively in a group, are asked to predict the height from which the cat should be dropped so that it will land on its feet if initially its back is pointing down. The calculation is challenging in that it requires that the students bring together a number of concepts and cannot be solved by simple "plug and chug" techniques.
Starting with the cat's parameters -- its dimensions, its mass and mass distribution, and the rate of rotation of the tail -- students must calculate the moments of inertia of the tail and of the body, then the angular momentum of both, and thus the angular velocity of the body. Finally they need to relate the angular velocity to the fall time, which then gives them the desired height.
Our cat has a cylindrical tail, which rotates about one end so that the moment of inertia is easily calculated from Itail =1/3 ml2. It is attached to a larger "solid" cylindrical body. While the tail is fabricated from solid brass with uniform density, the body contains the motor to drive the tail, batteries, a switching mechanism, and has legs attached. Thus it is anything but uniform. To keep the calculation simple, the students are told to assume the body is solid and uniform. They are given an effective mass M for the body so that the equation Ibody=1/2 Mr2 gives a value equal to the experimental moment of inertia.
Various materials were considered and tried for our construction. The current version, although we make no claims that we have uncovered the best combination, consists of a 7" piece of 4" diameter plastic drain tile as the body with plastic end caps. This produces an overall body dimension of L= 21.5 cm, r= 5.5 cm and m=0.907 kg. The tail is a 12.5 cm piece of solid brass cylinder of 0.6 cm radius with a mass of 0.14 kg. The motor used to drive the tail is taken from a Capsela™ construction set (available in many toy stores.) We used a standard motor module from the set and a gear module to lower the angular velocity. 2 AA batteries power the motor. The tail was attached using Capsela™ components as well. Students assume a constant angular velocity for the tail; the motor is sufficiently strong that it appears to instantly accelerate to a constant angular velocity limited by the internal friction of the gear module.
The tail is started rotating by a photogate that is normally blocked by a small plastic card inserted into the body of the cat. The card pulls out at the moment the cat is dropped, which completes the circuit and starts the tail rotating. This insures that the cat has zero angular momentum before being dropped. A ¾ s timer circuit is included to shut off the tail at the end of the fall.
The legs are pieces of plastic tubing glued to bolts in the body. The legs unbalance the body and must be counterbalanced with appropriately located weights hidden inside. When all the components were in place the tube was filled with spray-in insulating foam to keep everything in place.
We will use the cat as a TA led collaborative group activity in a workshop or lab. The students will be given the cat's parameters and asked to predict the height from which the cat can fall and land on its feet. The TA will then poll the various groups to determine the range of predictions. Because our cat cannot reverse its tail's rotation to stop its body from rotating once it has reached an upright orientation, the height from which the cat must be dropped to land on its feet is very specific. If dropped from too low a height, it will not have rotated enough, or from too high, it rotates too much. In either case, the cat will land on its side. The TA will then drop the cat from the "consensus" prediction. Once the cat lands on its feet -- or crashes -- the TA will demonstrate that other predictions do not work and then lead a "post mortem" discussion of the mistakes that the unsuccessful groups made.
Our cat will have its first exposure to the classroom in the fall of 2001 semester so we cannot, as yet, provide a report on its effectiveness as a teaching tool. We expect that the students will enjoy the experience and find it memorable, enhancing their understanding of angular momentum conservation.
We gratefully acknowledge many useful discussions with Hsu-Chang Lu, which provided many valuable solutions to difficult details.