Trevor C Lipscombe

Complementary curves of descent

The shapes of two wires in a vertical plane with the same starting and ending points are described as complementary curves of descent if beads frictionlessly slide down both of them in the same time, starting from rest. Every analytic curve has a unique complement, except for a cycloid (solution of the brachistochrone problem), which is self complementary. A striking example is a straight wire whose complement is a lemniscate of Bernoulli. Alternatively, the wires can be tracks down which round objects undergo a rolling race. The level of presentation is appropriate for an intermediate undergraduate course in classical mechanics.

Oscillations of a quadratically damped pendulum

A physical pendulum consisting of a circular disk at the end of a thin metal rod is connected to a low-friction rotary motion sensor, so that its angular position and velocity can be accurately measured. The disk can be oriented either perpendicular or parallel to the plane of swing to give significant or negligible air drag, respectively. The motion is analytically modeled in phase space. A quadratic dependence of the damping torque on the angular velocity fits the results. This laboratory experiment is suitable for undergraduate physics majors taking a first or second course in classical mechanics.

Babylonian resistor networks

The ancient Babylonians had an iterative technique for numerically approximating the values of square roots. Their method can be physically implemented using series and parallel resistor networks. A recursive formula for the equivalent resistance Req is developed and converted into a nonrecursive solution for circuits using geometrically increasing numbers of identical resistors. As an example, 24 resistors R are assembled into a second-order network and Req/R is measured to equal to better than 0.2%, as could be done in an introductory physics laboratory.

On the run: Unexpected outcomes of random events

When purportedly random processes give rise to surprisingly nonrandom outcomes, how can one tell whether the process are truly random? One way exploits the relatively little-known method of runs. We discuss the perplexing probability problem that brought this procedure to our attention, and our experimental tests of random processes produced by (a) numerical algorithm, (b) coin selection, and (c) nuclear decay.

The Physics of Shot Towers

In the late 18th and throughout the 19th century, lead shot for muskets was prepared by use of a shot tower. Molten lead was poured from the top of a tower and, during its fall, the drops became spherical under the action of surface tension. In this article, we ask and answer the question: How does the size of the lead shot depend on the height of the tower? In the process, we explain the basic technology underlying an important historical invention (the shot tower) and use simple physics (Newtonian mechanics and the thermodynamic laws of cooling) to model its operation.

Levitating a strip of paper by blowing over it

It is shown that if you blow vigorously over a curved strip of paper, it levitates into the shape of a catenary. This result quantifies a common classroom demonstration and is a pedagogically useful addition to other studies of catenaries in an intermediate classical mechanics course.

The rain-powered cart

A frictionless cart in the shape of a right triangle (with the vertical side forward) is elastically impacted by vertically falling raindrops. The speed of the cart as a function of time can be analytically deduced as an exercise in the use of trigonometric and hyperbolic functions. A characteristic time defines the approach to a terminal speed which is a sizeable fraction of the speed of the rain. The treatment is accessible to a student in a calculus-based mechanics course.

Dropping a particle out of a roller coaster

A rider in a roller coaster lets go of a particle such as a small marble. How far does the marble travel horizontally from the point of release before hitting the ground, assuming the speed of the roller coaster is determined by conservation of mechanical energy starting from the highest hill up which it was pulled? Where should the marble be released along the track if one wishes to maximize the range of the marble? These questions constitute interesting variations on conventional problems in two-dimensional kinematics, appropriate for an undergraduate course in classical mechanics. Exploration of various shapes of tracks could form interesting student projects for numerical or experimental investigation.