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The Young-Laplace (Y-L) equation relates the pressure difference across the interface of two fluids (such as air and water) to the curvature of the interface. The pressure rises on crossing a convex interface such as a rain drop and falls on crossing a concave interface such as the meniscus of water in a glass capillary. The relation between surface geometry and pressure difference across the interface provides the key concept in understanding a large collection of phenomena in hydrostatics. Of the many phenomena in hydrostatics that can be explained readily by the application of the Y-L equation, we consider the ones that are of particular interest in the introductory physics courses. These include the differential pressure within soap bubbles and liquid droplets, the rise and fall of liquids in capillaries, and the depth of liquid spills.

The stately fall of magnets through conducting pipes is a favorite classroom and laboratory activity used in teaching electromagnetic induction, Lenz's law, eddy currents, electromagnetic braking, and even Newton's third law. When a neodymium magnet is used, the terminal velocity is reached in just a few milliseconds as the induced eddy currents brake the fall and limit the speed to just a few centimeters per second. When the pipe is whole, the induced eddy currents in the vicinity of the poles form the circumferential current loops ahead and behind the falling magnet. When a slit is cut along the entire length of the conducting pipe, the induced eddy currents near the poles follow a path along the cut slit to complete a closed double loop. In this case the braking action is less effective so the magnet falls with a greater terminal velocity to generate the required eddy currents to balance the gravitational force. To explore experimentally the braking action due to each of the two types of induced eddy current, the terminal velocity of two neodymium magnets is measured, one cylindrical and one spherical, each falling through two identical copper pipes, one of which has a cut slit. The experiments are repeated with another set of copper pipes with the same outer dimensions as the first set but with a thinner wall to explore the effect of wall thickness. This article describes the experimental data, equation of motion, and outcomes of the experiments.

With the wide availability of strong neodymium magnets, the slow and stately fall of a magnet through a conducting pipe has become a favorite classroom demo for teaching electromagnetic induction, Newton's third law, and Lenz's law. Since Lenz's law is conceptually difficult for some students, several authors have used this demo to explore in greater detail the role of eddy currents in braking the fall of the magnet through the conducting pipe. The demo also serves as an excellent application of Newton's third law in a clear case of action-at-a-distance interaction. Other authors have focused attention on various aspects of this demo, from methods for measuring the terminal velocity to detailed analysis of electromagnetic damping. In this paper we explore the equilibrium condition of the magnet as it falls with constant velocity through a conducting pipe and present two simple methods for measuring the electromagnetic braking force directly during the magnet's fall.

When a chain hangs loosely from its end points, it takes the familiar form known as the catenary. Power lines, clothes lines, and chain links are familiar examples of the catenary in everyday life. Nevertheless, the subject is conspicuously absent from current introductory physics and calculus courses. Even in upper-level physics and math courses, the catenary equation is usually introduced as an example of hyperbolic functions or discussed as an application of the calculus of variations. We present a new derivation of the catenary equation that is suitable for introductory physics and mathematics courses.

Many of the old demonstrations in electricity and magnetism can be performed more effectively and with greater visual appeal by using newly available neodymium magnets and color LEDs. For this reason several demonstrations, commonly used for teaching electromagnetic induction and Lenz's law, have received renewed attention in recent years. In particular strong neodymium magnets have been used in several new demonstrations and experiments to help students learn the subtleties of eddy currents and magnetic braking. Here we describe several demonstrations that use a strong neodymium magnet, a coil, and a few LEDs to teach students in introductory physics the fundamentals of electromagnetic induction. More importantly the addition of LEDs provides a visual demonstration of Lenz's law, with which many students have some difficulty. The demonstrations described here are quite simple to construct, fun to do, and visually appealing; they may be used either in the laboratory or in the classroom.

A chain assumes the well-known shape known as a catenary when it hangs loosely from two points in a gravitational field. The correct solution of the catenary was one of the early triumphs of the newly invented calculus of variations at the end of the 17th century. Here we revisit the catenary and show that, for a chain hanging from a horizontal rod, three new and distinct configurations are possible if a soap film covers the area bounded by the chain and the rod. We first review the general problem and discuss the conditions under which the chain assumes a concave, triangular or convex configuration. The deciding factor is the strength of surface tension relative to the gravitational force per unit length of the chain. The conditions under which the chain assumes the shape of a perfect triangle are discussed in greater detail and analysed to obtain the tension along the chain. The triangular configuration is especially intriguing to undergraduates and may be used as a simple experiment to obtain the surface tension of the soap solution by measuring just one angle of the triangle. (Contains 7 figures.)

Undergraduate physics majors typically begin their study of modern physics with special relativity. It is here that physics students first encounter the counterintuitive concepts of time dilation and length contraction. Unfortunately, the derivations of these results are often cloaked in several layers of analysis that render them rather mysterious and otherworldly for novice students. Here we show how a very simple thought experiment leads directly to a student-friendly derivation of time dilation and length contraction. The only assumption needed is the constancy of the speed of light.