In 1971 during the Apollo 15 mission, Commander David Scott performed the Hammer-Feather Experiment to demonstrate that in the absence of air, a feather (listed by NASA as 30 g but large feathers we have weighed range from 0.3 to 3 g) would drop to the ground as fast as a 1.3-kg hammer. This same experiment, of course, has been done in many vacuum chambers on earth.
Inspired by the 50th anniversary of the Apollo 11 landing, we have created such an exhibit to complement our wind tunnel exhibit. The apparatus and pictures of the objects used in the exhibit are shown in the following picture, and they have been used to compare the fall time in three situations: (1) free fall in a room, (2) guided fall in the air-filled tube, and (3) guided fall in the evacuated tube.
The starting point for understanding this experiment is the same as for the wind tunnel: gravitational force is proportional to an object’s mass times the acceleration of gravity, G:
Fg = Mass × G
The value of G at a planet’s or moon’s surface is proportional to the mass of the planet or moon and its radium. It is 9.81 m/s2 on Earth and 1.625 m/s2 on our moon. Even though the gravitation force is proportional to the object’s mass, the energy required to accelerate it to a given velocity is also, so all objects accelerate at the same rate if there is no air to push out of the way.
Solid balls of similar size to a baseball ranging from 143 to 8.3 g all take about 0.4 seconds to fall a meter. All fall times reported here are averages of 3-7 measurements, with an average standard deviation of 0.05 s. The calculated time for free fall is 0.45 s. The paper cylinders, with and without a bottom, often tumble and take about 0.5 s. The difference of 0.1 s corresponds to about 0.2 m (8 inches), so it is clearly visible when dropped next to each other.
We now consider guided fall in the air-filled tubes. The tubes are 0.91-m long and have an 8.9-cm internal diameter. The first set of experiments is with two hollow paper cylinders, both 6.7-cm in diameter, and three paper cylinders closed on one end with diameters of 5.4, 6.7, and 7.6 cm. The results are shown in the following figure:
The fall time of the open cylinders is a bit longer than the free fall time, and the fall time of the closed cylinders depends on its cross-sectional area. The closed cylinders extrapolate to the open cylinder result at a cross-sectional area of about 20 cm2. The interpretation of this data is more complicated than a simple drag factor, however, because the annular area between the tube and the cylinder gets smaller as the cylinder gets larger.
The next experiment is with an ordinary Styrofoam cup with various weights inside ranging from 0.7 to 12.2 g. As shown in the following figure, the drop time gets smaller as the weight increases (at constant cross-sectional area, of course). The heaviest weight gives a fall time about the same as the open paper cylinder.
Correlating this data with other miscellaneous shapes and weights is a challenge. However, from the previous two figures one might imagine that the fall time is proportional to cross-sectional area and inversely proportional to mass. That works to some degree, but the correlation works better if one also considers the annular area. The conductance of the air around the object increases with the annular area. The best example of the resistance of the annulus comes from comparing the 0.66 s drop time of the Styrofoam ball in the tube to 0.45 s in open air.
The mathematical reasoning now becomes a bit more complicated. We want a function that goes to unity (no effect) when the annulus goes to infinity and goes to infinity (infinite drop time) when the annulus goes to zero. Such a function is (annular fraction + 1)/annular fraction. The resulting plot is far better than simply correlating with object weight and cross-sectional area, but it does not capture all the physics. Also recall that the average standard deviation of the times is 0.05 s, so some of the scatter is experimental error.
At the end of the day, all this complication goes away when the tubes are evacuated. The drop times become equal within the limits of human perception at about 0.45 s.