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To understand the concept of spring constant, we begin with Hooke's Law. This fundamental law in physics states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, this can be expressed as \[ F = -kx \] where:

• \( F \) is the force applied to the spring,
• \( k \) is the spring constant, and
• \( x \) is the displacement of the spring from its natural length.

The spring constant \( k \) is a measure of the stiffness of a spring. A higher \( k \) indicates a stiffer spring that requires more force to achieve the same displacement. In the context of the given problem, we calculated the spring constant by first using the empty chair's mass and oscillation period. Knowing \( m = 42.5 \, \text{kg} \) and \( T = 1.3 \, \text{s} \), we solved the equation: \[ T = 2 \pi \sqrt{\frac{m}{k}} \] Rearranging for \( k \), the spring constant is approximately \( 13.6 \, \text{N/m} \). This helped us proceed to find the astronaut's mass.

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement. It is one of the most important principles for understanding oscillations. The detailed features of SHM include:

• The motion is repetitive with a fixed rhythm.
• The force driving the object towards the center is proportional to the displacement.
• The energy in SHM is conserved and continuously switches between kinetic and potential forms.

In the given problem, the oscillations of the chair and astronaut follow simple harmonic motion. The essential formula used is the period of oscillation \( T \), given by: \[ T = 2 \pi \sqrt{\frac{m}{k}} \] This signifies that the period is dependent on both the mass \( m \) and the spring constant \( k \). When the chair is occupied by the astronaut, the longer period implies increased effective mass, which tells us about the mass of the astronaut when solved.

In space, conventional methods of weighing do not work due to the lack of gravity. Instead, astronauts’ mass is measured using principles like those involving oscillations. The key idea is that a known spring constant can be used to determine an unknown mass based on the period of oscillation.Using the known values of the empty chair, the spring constant was first identified. As this same constant is applicable when the astronaut joins the system, her mass was calculated from the altered period. The derived formula for finding the astronaut's mass \( m \) is \[ m = \left( \frac{T}{2 \pi} \right)^2 \cdot k \] where \( T \) is the new period with the astronaut. Upon performing the calculations, the astronaut’s mass works out to approximately \( 130 \, \text{kg} \), reflecting the simplicity and efficiency of using oscillations for mass measurement in a zero-gravity environment.