91Ó°ÊÓ

14\. BMMD Astronauts sometimes use a device called a body-mass measuring device (BMMD). Designed for use on orbiting space vehicles, its purpose is to allow astronauts to measure their mass in the "weightless" conditions in Earth orbit. The BMMD is a springmounted chair; an astronaut measures his or her period of oscillation in the chair; the mass follows from the formula for the period of an oscillating block-spring system. (a) If \(M\) is the mass of the astronaut and \(m\) the effective mass of that part of the BMMD that also oscillates, show that $$ M=\left(k / 4 \pi^{2}\right) T^{2}-m $$ where \(T\) is the period of oscillation and \(k\) is the spring constant. (b) The spring constant was \(k=605.6 \mathrm{~N} / \mathrm{m}\) for the BMMD on Skylab Mission Two; the period of oscillation of the empty chair was \(0.90149 \mathrm{~s}\). Calculate the effective mass of the chair. (c) With an astronaut in the chair, the period of oscillation became \(2.08832 \mathrm{~s}\). Calculate the mass of the astronaut.

Step by step solution

01

Understanding oscillation and mass relationship

The formula for the period of an oscillating block-spring system is given by: \[T = 2\pi \sqrt{\frac{M}{k}}\] Where \(T\) is the period, \(M\) is the mass, and \(k\) is the spring constant.

02

Rearranging the period formula

Square both sides of the equation: \[T^2 = (2\pi)^2 \frac{M}{k}\] This simplifies to: \[T^2 = \frac{4\pi^2 M}{k}\]

03

Solving for the mass

Solving for \(M\), we rearrange the equation to get: \[M = \frac{k T^2}{4\pi^2} - m\] where \(m\) is the effective mass of that part of the BMMD that also oscillates.

04

Calculating effective mass of the chair

Given \(k = 605.6 \ \text{N/m}\) and \(T = 0.90149 \ \text{s}\) for the empty chair, use the formula: \[M = \frac{k T^2}{4\pi^2} - m\] Set \(M = 0\) and solve for \(m\). \[0 = \frac{605.6 \cdot (0.90149)^2}{4\pi^2} - m\] Therefore, \[m = \frac{605.6 \cdot (0.90149)^2}{4\pi^2}\] This gives: \[m = 12.31 \ \text{kg}\]

05

Calculating mass of the astronaut

With an astronaut in the chair (new period \(T' = 2.08832 \ \text{s}\)), we use the same formula: \[M + m = \frac{605.6 \cdot (2.08832)^2}{4\pi^2}\] Using previously calculated \(m = 12.31 \ \text{kg}\): \[M = \frac{605.6 \cdot (2.08832)^2}{4\pi^2} - 12.31 \] Calculate: \[M = 20.80 \ \text{kg}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Oscillation Period

Understanding the concept of oscillation period is key for solving mass measurement problems in space. The oscillation period, denoted as T, is the time it takes for an object to complete one full cycle of motion. For example, in the context of the Body-Mass Measuring Device (BMMD) used by astronauts, the period is the time the spring-mounted chair takes to move back and forth once. A longer period implies a slower oscillation, whereas a shorter period indicates a quicker oscillation.
To calculate the period of a block-spring system, the formula is:
\( T = 2\pi \sqrt{\frac{M}{k}} \)
Where:

• \(T\) represents the period.
• \(M\) is the mass of the object.
• \(k\) is the spring constant.

This formula helps understand the relationship between mass and period, which is fundamental in the BMMD scenario for determining an astronaut's mass in a weightless environment.

Block-Spring System

The oscillation of a block-spring system is a classical physics problem that directly relates to how the BMMD works. In this system, a block is attached to a spring, and when displaced, it oscillates back and forth around an equilibrium position. The system's behavior is governed by Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position:
\( F = -kx \)
Where:

• \(F\) is the force exerted by the spring.
• \(k\) is the spring constant.
• \(x\) is the displacement from equilibrium.

The oscillation period can be derived from this relationship and, importantly, it allows us to measure mass in space. By knowing the period of oscillation and the spring constant, we can determine the mass of the oscillating system using the provided formula:
\[ M = \left( \frac{k T^{2}}{4 \pi^{2}} \right) - m \]
This equation helps calculate the effective mass (\(m\)) and, more importantly, can be used by astronauts to measure their mass accurately in microgravity environments.

Spring Constant

The spring constant (\(k\)) is a crucial parameter in understanding the dynamics of oscillating systems. It measures the stiffness of a spring; a higher spring constant means a stiffer spring that is harder to compress or stretch. In the BMMD, knowing the spring constant is essential because it directly influences the oscillation period.
Hooke's Law, \( F = -kx \), connects the force applied to the spring and the displacement. The constant \(k\) also appears in the period formula for a block-spring system:
\( T = 2\pi \sqrt{\frac{M}{k}} \)
In our exercise, the spring constant for the BMMD is given as \(k = 605.6 \, \text{N/m}\).
This means for every meter the spring is compressed or stretched, it exerts a force of 605.6 newtons. By using this value along with the measured period, we can accurately determine the mass of an object or an astronaut in space.

Effective Mass

Effective mass (\(m\)) in the context of the BMMD refers to the mass of all parts of the device that contribute to its oscillation. This includes the chair and any other parts that move with the astronaut. The effective mass must be accounted for separately from the astronaut's mass when calculating the oscillation period.
In our problem:

• We first need to find the effective mass by observing the oscillation period of the empty chair.

Given the spring constant \(k = 605.6 \, \text{N/m}\) and the period of the empty chair \(T = 0.90149 \, \text{s}\), we calculate:
\[ m = \frac{605.6 \cdot (0.90149)^{2}}{4\pi^{2}} = 12.31 \, \text{kg} \]
Once we have \(m\), we can compute the astronaut's mass by using the new period with the astronaut in the chair. For example, if the new period \(T'\) is 2.08832 seconds, the mass of the astronaut (\(M\)) can be found using:
\[ M = \frac{605.6 \cdot (2.08832)^{2}}{4\pi^{2}} - 12.31 \]
This method ensures we get accurate measurements, even in the challenging conditions of space.