Astronauts who spend long periods of time in orbit periodically measure their body masses as part of their health-maintenance programs. On earth, it is simple to measure body weight W with a scale and convert it to mass m using the acceleration due to gravity, since W=mg. However, this procedure does not work in orbit, because both the scale and the astronaut are in free-fall and cannot press against each other (see Conceptual Example 12 in Chapter 5). Instead, astronauts use a body mass measurement device, as Figure 10.15 illustrates. This device consists of a spring-mounted chair in which the astronaut sits. The chair is then started oscillating in simple harmonic motion. The period of the motion is measured electronically and is automatically converted into a value of the astronaut’s mass, after the mass of the chair is taken into account. The spring used in one such device has a spring constant of 606 N/m, and the mass of the chair is 12.0 kg. The measured oscillation period is 2.41 s. Find the mass of the astronaut.

Figure 10.15  Astronaut Tamara Jernigan uses a body mass measurement device to measure her mass while in orbit. (Courtesy NASA.)

Reasoning  The relation (Equation 10.11) can be solved for the mass m in terms of the spring constant k and the angular frequency w. The spring constant is known. Although the angular frequency is not known, it can be related to the given oscillation period of T=2.41 s by using w=2p/T (Equation 10.4). The mass calculated using Equation 10.11 is the total mass of the astronaut and the chair, so that it will be necessary to subtract the mass of the chair to obtain the mass of the astronaut.

Solution Using Equations 10.11 and 10.4 gives