Let's break down your question about the Body Mass Measurement Device (BMMD) and how it functions in a weightless environment to measure an astronaut's mass. We will approach this problem in parts as you've outlined: first, we’ll derive the relevant formulas, then we’ll calculate the effective mass of the chair, and finally, we’ll determine the astronaut's mass.

Understanding the Oscillation of the BMMD

The BMMD operates based on the principles of simple harmonic motion (SHM). When a mass is attached to a spring, it oscillates back and forth. The key relationship that allows us to connect the period of oscillation (T), the mass (M), and the spring constant (k) is given by the formula:

T = 2π√((M + m)/k)

In this formula, T is the period of oscillation, M is the mass of the astronaut, m is the effective mass of the chair, and k is the spring constant. We can deduce that the total mass affecting the oscillation is the sum of the astronaut's mass and the effective mass of the chair.

Part (a): Deriving the Relationship

To show that the formula holds, we start from the basic principles of oscillation. For a spring-mass system, the period of oscillation depends on both the mass of the system and the spring constant. When we look at the effective mass, we consider that the chair itself has a mass that contributes to the overall oscillation. Thus, we can express the period of oscillation as:

T^2 = 4π^2((M + m)/k)

Rearranging this gives us:

M + m = (k * T^2) / (4π^2)

This demonstrates how the astronaut's mass and the effective mass of the chair relate to the oscillation period and the spring constant.

Part (b): Calculating the Effective Mass of the Chair

Now, let’s calculate the effective mass of the chair using the given values. We know:

• k = 605.6 N/m
• T (period of empty chair) = 0.90149 s

First, we will substitute T into the rearranged formula:

M + m = (605.6 N/m * (0.90149 s)^2) / (4π^2)

Calculating the right side:

(0.90149 s)^2 ≈ 0.81268 s²

M + m = (605.6 * 0.81268) / (4 * π²)

Calculating further:

4 * π² ≈ 39.478

M + m ≈ (605.6 * 0.81268) / 39.478

Calculating this gives:

M + m ≈ 12.18 kg

Now, we can conclude that the effective mass of the chair (m) can be determined. If we assume the astronaut's mass (M) is negligible for this part, then:

m ≈ 12.18 kg

Part (c): Determining the Mass of the Astronaut

Now, with the astronaut sitting in the chair, we have a new period of oscillation:

• T (with astronaut) = 2.08832 s

We will again use the same formula:

M + m = (k * T^2) / (4π^2)

Substituting the known values:

M + 12.18 kg = (605.6 N/m * (2.08832 s)^2) / (4π^2)

Calculating (2.08832 s)^2 ≈ 4.3627 s²:

M + 12.18 kg = (605.6 * 4.3627) / 39.478

This will give:

M + 12.18 kg ≈ 69.59 kg

Solving for M:

M ≈ 69.59 kg - 12.18 kg = 57.41 kg

Thus, the mass of the astronaut is approximately 57.41 kg. This method effectively illustrates how the principles of oscillation can be applied to measure mass in a microgravity environment, showcasing the clever design of the BMMD.