USIYPT 2018: Problems

1. The Moon’s Orbit

The great astronomer Hipparchus was born in Nicaea in the same year of a total solar eclipse. Later he used drawings of this eclipse, as observed from different locations at the same time, to accurately measure the distance to the moon. Eighteen centuries later, Johannes Kepler mathematically characterized the moon’s orbit using his first two laws of planetary motion. From this time on, the orbit of the moon has been extremely well understood.

In this problem you must measure the moon’s three main orbital elements: (1) the period of its orbit, (2) the eccentricity of its orbit, and (3) the semi-major axis of its orbit.

2. Electromagnetically Coupled Mechanical Oscillators

An interesting problem in classical mechanics is that of weakly coupled mechanical oscillators. One common example is a playground swing-set, where pendulums are attached to a common bar.

For this problem you must set-up a system of coupled mechanical oscillators, but rather than coupling them using a mechanical device, such as a rod, they must be coupled electromagnetically without any external power source. How does your system work fundamentally?

3. Projectile Motion through Air

Pitchers throw curveballs, tennis players hit with topspin, and soccer strikers bend their shots. These all contradict Galileo’s simple model of constant downward acceleration.

For this problem, you are to investigate the motion of a spinning ball theoretically and experimentally. How, and under what circumstances, does the Galilean model need to be adapted?

4. Incandescent Light Bulbs and Blackbody Radiation

A common incandescent light bulb works by electrically heating a filament until it radiates visible light.   Until Max Planck unified them in 1900, there were two different proportionality laws relating the emitted light as a function of the surface temperature of a particular filament.   These were: (1) Stefan’s Law for finding the total power radiated (commonly called wattage), and (2) Wien’s Law for finding the wavelength of the maximum spectral flux density (commonly called color), which can be written, respectively, as follows:

\left( 1 \right)\ P\propto {{T}^{^{4}}},\quad \left( 2 \right)\,{{\lambda }_{\max }}\propto \frac{1}{T}\,.

For this problem, use common incandescent light bulbs to verify the proportionality laws of both Stefan and Wien.