A sundial in Rio de Janeiro
- Round
- Theory
- Points
- Not given
- Level
- Silver
- Taught in
- Section 1.7
Adapted from IOAA 2012 (Rio de Janeiro), theory problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 15 October 2026.
Above the door of the dome of the 32 cm telescope at Brazil's National Observatory, in Rio de Janeiro (22°54′ S, 43°12′ W), there is a sundial facing north. Its dial lies in the vertical plane through the east point, the zenith and the west point, and its rod is parallel to Earth's axis. For which declinations of the Sun, and during which part of the year (months and seasons), does the sundial (i) fail to work for at least part of the day, and (ii) not work at all during the day?
- The dial faces north, so the Sun lights it, and the rod can cast a shadow on it, only while the Sun is north of the vertical plane through E, Z and W, the prime vertical (Section 1.2).
- The Sun rises north of east and sets north of west when its declination is positive, and south of them when it is negative (Section 1.3). So when
- The Sun crosses the meridian north of the zenith only if its declination is more than the declination of the zenith, which is the latitude, −22°54′ ((1.8)). If
Answer(i) The sundial fails near sunrise and sunset whenever the Sun's declination is negative, from the September equinox to the March equinox, in the southern spring and summer. (ii) It does not work at all when the declination is −22°54′ or less, for about three weeks around the December solstice, which is the southern summer solstice.
This problem also appears among the olympiad problems of Section 1.7, Problems for Chapter 1, which teaches the ideas it uses. Equation and figure numbers refer to the course.