Problem 40 - College Physics

Question

Weighing astronauts. In order to study the long-term effects of weightlessness, astronauts in space must be weighed (or at least "massed"). One way in which this is done is to seat them in a chair of known mass attached to a spring of known force constant and measure the period of the oscillations of this system. If the \(35.4 \mathrm{~kg}\) chair alone oscillates with a period of \(1.25 \mathrm{~s},\) and the period with the astronaut sitting in the chair is \(2.23 \mathrm{~s},\) find (a) the force constant of the spring and (b) the mass of the astronaut.

Step-by-Step Solution

Answer

The spring constant is approximately \(1786.4\; \mathrm{N/m}\), and the astronaut's mass is approximately \(103.94\; \mathrm{kg}\).

Step 1: Understanding the Problem

We are given a chair with a known mass of \(35.4\; \mathrm{kg}\) and its oscillation period \(T_c = 1.25\; \mathrm{s}\). With the astronaut sitting on the chair, the period is \(T_{a+c} = 2.23\; \mathrm{s}\). Our goal is to find the spring constant \(k\) and the astronaut's mass \(m_a\).

Step 2: Relate Period and Spring Constant

The formula for the period of oscillation 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.

Step 3: Calculate Spring Constant

For the chair alone, we have \(T_c = 1.25\; \mathrm{s}\) and \(m_c = 35.4\; \mathrm{kg}\). Using the period formula: \[ 1.25 = 2\pi \sqrt{\frac{35.4}{k}} \] Squaring both sides, we get: \[ k = \frac{4\pi^2 \times 35.4}{1.25^2} \approx 1786.4\; \mathrm{N/m} \] by solving for \(k\).

Step 4: Calculate Total Mass with Astronaut

With the astronaut in the chair, we have \(T_{a+c} = 2.23\; \mathrm{s}\). Using the same formula: \[ 2.23 = 2\pi \sqrt{\frac{m_{a+c}}{k}} \] Squaring both sides, we find: \[ m_{a+c} = \frac{4\pi^2 k}{2.23^2} \] Substituting the value of \(k\) we found: \[ m_{a+c} \approx \frac{4\pi^2 \times 1786.4}{2.23^2} \approx 139.34\; \mathrm{kg} \]

Step 5: Determine Astronaut's Mass

Knowing the total mass \(m_{a+c}\) and the chair's mass \(m_c\), the astronaut's mass is calculated by: \(m_a = m_{a+c} - m_c\). Thus, \[ m_a = 139.34 - 35.4 \approx 103.94\; \mathrm{kg} \].

Key Concepts

Mass-Spring System

A mass-spring system consists of a mass attached to a spring that can compress or stretch. This simple system oscillates back and forth when set into motion. Such movement is a classic example of mechanical oscillations.
The main characteristic of this system is its periodic motion, meaning it repeats after a set period. This system's natural vibrational response results from the force exerted by the spring returning it to a position of equilibrium whenever it is disturbed.
In our space experiment, the chair acts as the mass and when an astronaut sits in it, the total mass increases. The combination of a known mass (the chair) and a variable mass (the astronaut) allows us to study how added mass affects oscillations.

Spring Constant Calculation

The spring constant, symbolized as \( k \), indicates how stiff the spring is. It measures the force needed to compress or extend the spring by a specific distance.
With a simple formula for oscillation period \( T \) in a mass-spring system, shown as \( T = 2\pi \sqrt{\frac{m}{k}} \), we can uncover \( k \).
The oscillation period is directly connected to both the spring constant and the mass of the object. By rearranging this equation, we derived the spring constant as:

• \( k = \frac{4\pi^2 \times m}{T^2} \)

The spring constant \( k \) reveals important information on how the spring behaves with different weights. In our scenario, calculating \( k \) using the known mass and original oscillation period helps in further measurements.

Harmonic Motion in Space

Harmonic motion occurs when the force on an object is proportional and opposite to its displacement from equilibrium. This results in repetitive oscillations.
In space, where gravity doesn't influence the oscillation as much as it does on Earth, harmonic motion helps in determining unknown masses, like that of an astronaut.
Space offers a unique setting where the principles of harmonic motion extend to experiments in long-term weightlessness. Here, measurements rely on tools like the mass-spring system for accuracy as springs provide restorative force to counteract initial displacement. Using harmonic motion laws allows precise evaluations of how systems behave in environments different from Earth.

Astronaut Mass Measurement

In the weightlessness of space, traditional weighing scales become ineffective. Instead, oscillation periods of a mass-spring system allow for mass measurement.
By seating the astronaut in a known mass chair and noting the oscillation period, we can determine their mass. Initially, by finding the spring constant using the chair's mass and oscillation period, the combined period provides a complete mass total when the astronaut is added.
The formula rearranges to solve: the adjusted period lets us calculate a total mass, which includes both the astronaut and the chair. The astronaut's mass is then derived by subtracting the chair's mass from this total.
This method ensures accurate mass readings in zero-gravity conditions, crucial for health monitoring and equipment calibration.