Baron von Münchausen and Intergalactic Travel

Recently found archives of the late Baron von Münchausen brought to light some unpublished scientific papers. In one of them, his calculations indicated that the Sun’s energy would some day be exhausted, with the subsequent freezing of the Earth and its inhabitants. In order to avert this inevitable outcome, he proposed the construction of a large, rigid balloon, empty of all gases, 1 km in radius, and attached to the Earth by a long, light

Figure 1.1

rope of extreme tensile strength. The Earth would be propelled through space to the nearest star via the Archimedes’ force on the balloon, transmitted through the rope to the large staple embedded in suitable bedrock (see Figure 1.1). Estimate the force on the rope (assuming a massless balloon). Discuss the feasibility of the Baron’s idea (without using any general statements).

SOLUTION

The general statement that a closed system cannot accelerate as a whole in the absence of external forces is not usually persuasive to determined inventors. In this case, he would make the point that the force on the rope is real. To get an estimate of this force, assume that the balloon is just above the surface of the Earth and that the density of air is approximately constant to 2 km. Archimedes tells us that the force on the rope will equal the weight of the air, mass m, excluded by the empty balloon (given a massless balloon material). We then may use the ideal gas law

to find the force F_a:

Figure 1.2

Where we approximate the acceleration due to gravity as constant up to 2 km; i.e., g ~ 10 m/s². However, there will be no force acting on the Earth. The system (Earth + surrounding air) is no longer symmetric (see Figure 1.2a). The symmetric system would be the one with no air on the opposite side of the Earth (see Figure 1.2b). Therefore, there will be a force between this additional air, which can be treated as a “negative” mass, and the Earth (see Figure 1.2c):

where M_E and R_E are the mass and radius of the Earth, respectively, and G is the gravitational constant. So, the Archimedes force is completely canceled by the gravitational force from the air. Perhaps that is why the Baron shelved his idea.