Equilibria and Stability

Introduction

When there is zero net force and net torque on an object, its momentum and angular momentum are constant, and it is said to be in mechanical equilibrium. What if the object experiences a slight push such that it is no longer in perfect equilibrium? Will the imbalance in net force or torque return the object its equilibrium position, or cause it to move even farther out of equilibrium? This is an important question, because it is usually impossible to keep an object in perfect equilibrium. Stability is the resistance of an equilibrium position to small disturbances.

Materials

Pencil

Hardcover textbook

Circular washer magnets

Assorted collection of different magnets

Levitron

Methods

1. Try balancing a hardcover textbook on its bottom side. If you succeed, the book will be in equilibrium and will remain standing. Is this equilibrium stable? To find out, give the book's cover a gentle push near the top. See if you can push it gently enough that it rocks back and forth a bit but remains standing. If you can, that equilibrium is considered stable.

2. There are different degrees of stability. Repeat the above experiment with books of different thicknesses and heights, and try to figure out the relationship between these variables and how hard you must push before a book falls over. Stability can also vary with dimension: instead of pushing on the cover, try pushing on a book's left or right edge.

3. Now, try balancing the pencil on its tip. In theory, the pencil's center of mass could be positioned exactly above the tip such that the torque of gravity on the pencil is zero. This would be an equilibrium position for the pencil. The difficulty (impossibility) of balancing a sharp pencil on its tip indicates the instability of that equilibrium position: even a tiny deviation from the equilibrium position results in a torque that rotates the pencil even farther away from equilibrium.

4. Magnets exert forces on each other. Is it possible to levitate an object (that is, keep it in stable equilbrium in midair) by countering the force of gravity with magnetic attraction or repulsion? Try to levitate a magnet using nothing except other magnets.

5. Keep trying to levitate a magnet, but now use things besides magnets. For example, if you hold a pencil upright and thread it through the holes of several circular washer magnets, you can make some of them float above the table. In this case, the contact force between the pencil and the inside surface of the washer magnetic is a non-magnetic force.

6. If one is clever and resourceful enough, it is possible to create a system that levitates a magnet using only magnetic forces. The Levitron is such a system. Watch it go! However, note that this system is not technically in mechanical equilibrium, since the top is spinning.

Discussion Questions

1. When architects and engineers design a building, they must design it so that it stands up in stable equilibrium. What are some forces that tend to push buildings away from equilibrium?

2. Is a weight hanging down on the end of a string in stable or unstable equilibrium?

3. Is it harder to hit a golf ball into a hole at the top of a hill, or a hole at the bottom of a valley?

4. Can you think of other examples of stable and unstable equilibria?