Weighing Astronauts. This procedure has been used to 'weigh' astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

Verified step by step guidance

1. First, understand that the problem involves a mass-spring system where the period of oscillation is related to the mass attached to the spring. The formula for the period \( T \) of a mass-spring system is given by: \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( m \) is the mass and \( k \) is the spring constant.
2. Calculate the spring constant \( k \) using the period of the empty chair. Rearrange the formula to solve for \( k \): \( k = \frac{4\pi^2 m}{T^2} \). Substitute \( m = 42.5 \text{ kg} \) and \( T = 1.30 \text{ s} \) to find \( k \).
3. Next, use the period of the chair with the astronaut to find the total mass \( m_{total} \) (chair plus astronaut). Use the same formula for the period: \( T = 2\pi \sqrt{\frac{m_{total}}{k}} \). Substitute \( T = 2.54 \text{ s} \) and the previously calculated \( k \) to solve for \( m_{total} \).
4. The total mass \( m_{total} \) is the sum of the mass of the chair and the mass of the astronaut. Therefore, \( m_{total} = m_{chair} + m_{astronaut} \). Rearrange this to solve for the mass of the astronaut: \( m_{astronaut} = m_{total} - m_{chair} \).
5. Substitute the known values into the equation to find the mass of the astronaut. Use the calculated \( m_{total} \) and the given \( m_{chair} = 42.5 \text{ kg} \) to find \( m_{astronaut} \).

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory motion of objects like springs and pendulums. It is characterized by a restoring force proportional to the displacement from equilibrium, resulting in sinusoidal oscillations. The period of oscillation depends on the mass and the spring constant, which is crucial for determining the mass of the astronaut in this scenario.

Spring Constant

The spring constant, denoted as k, measures the stiffness of a spring and is defined by Hooke's Law, F = -kx, where F is the force applied, and x is the displacement. In the context of oscillations, the spring constant influences the period of vibration, allowing us to relate the change in period to the mass of the astronaut.

Period of Oscillation

The period of oscillation is the time taken for one complete cycle of motion. It is determined by the formula T = 2π√(m/k), where m is the mass and k is the spring constant. By comparing the periods with and without the astronaut, we can calculate her mass using the change in oscillation period.