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Chapter 11: Problem 38

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.

Short Answer

Expert verified
The spring constant is approximately 897.3 N/m, and the astronaut's mass is approximately 79.1 kg.

Step by step solution

01

Identify the formulas involved

The formula for the period of oscillation of a mass-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. We will use this formula to find the spring constant \( k \) and then the astronaut's mass.
02

Solve for the spring constant (k)

Using the chair's mass \( m = 35.4 \text{ kg} \) and its period \( T = 1.25 \text{ s} \), we find the spring constant \( k \):\[T = 2\pi \sqrt{\frac{m}{k}} \Rightarrow k = \frac{4\pi^2 m}{T^2}\]Substitute the given values:\[k = \frac{4\pi^2 \times 35.4}{(1.25)^2} \approx 897.3 \text{ N/m}\]
03

Solve for the mass of the astronaut

We know the total system (chair + astronaut) has a period of 2.23 s. Using\[T = 2\pi \sqrt{\frac{m + m_a}{k}}\]Substitute the known values:\[2.23 = 2\pi \sqrt{\frac{35.4 + m_a}{897.3}}\]Square both sides and solve for \( m_a \):\[\frac{(2.23)^2}{4\pi^2} \times 897.3 = 35.4 + m_a\]\[m_a = \frac{(2.23)^2}{4\pi^2} \times 897.3 - 35.4\]\[m_a \approx 79.1 \text{ kg}\]

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Key Concepts

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

Mass-Spring System
A mass-spring system is an important concept in physics, often used to study oscillations and harmonious motion. In such a system, a mass is attached to a spring, which can compress or stretch. The system oscillates back and forth, due to the restoring force of the spring. This motion is predictable and can be used to analyze various properties of the system, such as frequency and period of oscillation.

In the case of our exercise, the astronaut and the chair form a mass-spring system. The spring provides a known force that resists the motion of the mass when it is displaced from its equilibrium position. The characteristics of the system allow scientists to conduct various experiments, such as determining the mass of an object in space. This type of system is governed by Hooke's Law:
  • The force exerted by the spring is proportional to its displacement.
  • The formula is: \( F = -kx \) where:
    • \( F \) is the force.
    • \( x \) is the displacement.
    • \( k \) is the spring constant.
Spring Constant Calculation
The spring constant, denoted by \( k \), is a measure of the stiffness of a spring. It is crucial to determine this constant to predict how the system behaves when displaced. The spring constant can be calculated using the period of oscillation and the mass in the system. For a chair on a spring with a mass of 35.4 kg, the formula involved is derived from the basic formula of oscillation periods.

To calculate the spring constant, we use:
  • Formula: \( T = 2\pi \sqrt{\frac{m}{k}} \)
  • Reorganizing gives us: \( k = \frac{4\pi^2 m}{T^2} \)
Substituting the known values, the spring constant of the chair ends up being approximately 897.3 N/m. This means the spring is quite strong, capable of supporting the chair's weight and producing consistent oscillations.
Period of Oscillation
The period of oscillation is an essential concept when analyzing mass-spring systems, especially in zero gravity environments like space. The period is the time taken for one complete cycle of oscillation. It鈥檚 dependent on both the mass and the spring constant:

  • The main formula is: \( T = 2\pi \sqrt{\frac{m}{k}} \)
In our example, the period of the chair alone is 1.25 seconds, while with the astronaut, it is increased to 2.23 seconds. This change in period is primarily due to the added mass, which increases the system's inertia and results in slower oscillations. Understanding and measuring the period of oscillation can help determine unknown parameters in a system, like mass, especially in environments where traditional methods cannot be applied.
Mass Measurement in Space
Measuring mass in space is challenging because traditional scales rely on gravity. However, oscillation-based methods provide a clever solution. By examining how a known system oscillates with different masses, we can deduce the mass of an object, even in a weightless environment.

Astronauts measuring their mass while seated in a spring-attached chair are an excellent real-world application of this technique. When using this method:
  • The known variable is the spring constant.
  • Calculated variables are the period of oscillation and the resulting force.
  • The unknown is the mass of the astronaut, which is calculated using the system's changed period.
By considering the change in the system's period with the addition of the astronaut, we can accurately find their mass. This clever use of physics principles allows us to perform essential tasks outside Earth鈥檚 gravitational pull.

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Most popular questions from this chapter

Achilles tendon. The Achilles tendon, which connects the calf muscles to the heel, is the thickest and strongest tendon in the body. In extreme activities, such as sprinting, it can be subjected to forces as high as 13 times a person's weight. According to one set of experiments, the average area of the Achilles tendon is \(78.1 \mathrm{~mm}^{2}\), its average length is \(25 \mathrm{~cm},\) and its average Young's modulus is 1474 MPa. (a) How much tensile stress is required to stretch this muscle by \(5.0 \%\) of its length? (b) If we model the tendon as a spring, what is its force constant? (c) If a \(75 \mathrm{~kg}\) sprinter exerts a force of 13 times his weight on his Achilles tendon, by how much will it stretch?

The effect of jogging on the knees. High-impact activities such as jogging can cause considerable damage to the cartilage at the knee joints. Peak loads on each knee can be eight times body weight during jogging. The bones at the knee are separated by cartilage called the medial and lateral meniscus. Although it varies considerably, the force at impact acts over approximately \(10 \mathrm{~cm}^{2}\) of this cartilage. Human cartilage has a Young's modulus of about 24 MPa (although that also varies). (a) By what percent does the peak load impact of jogging compress the knee cartilage of a \(75 \mathrm{~kg}\) person? (b) What would be the percentage for a lower-impact activity, such as power walking, for which the peak load is about four times body weight?

"Seeing" surfaces at the nanoscale. One technique for making images of surfaces at the nanometer scale, including membranes and biomolecules, is dynamic atomic force microscopy. In this technique, a small tip is attached to a cantilever, which is a flexible, rectangular slab supported at one end, like a diving board (Figure 11.40 ). The cantilever vibrates, so the tip moves up and down in simple harmonic motion. In one mode of operation, the resonant frequency for a cantilever with force constant \(k=1000 \mathrm{~N} / \mathrm{m}\) is \(100 \mathrm{kHz}\). As the oscillating tip is brought within a few nanometers of the surface of a sample (as shown in the figure), it experiences an attractive force from the surface. For an oscillation with a small amplitude (typically \(0.050 \mathrm{nm}),\) the force \(F\) that the sample surface exerts on the tip varies linearly with the displacement \(x\) of the tip, \(|F|=k_{\text {surf }} x,\) where \(k_{\text {surf }}\) is the effective force constant for this force. The net force on the tip is therefore \(\left(k+k_{\text {surf }}\right) x,\) and the frequency of the oscillation changes slightly due to the interaction with the surface. Measurements of the frequency as the tip moves over different parts of the sample's surface can provide information about the sample. If we model the vibrating system as a mass on a spring, what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface? A. \(25 \mathrm{ng}\) B. \(100 \mathrm{ng}\) C. \(25 \mu \mathrm{g}\) D. \(100 \mu \mathrm{g}\)

An astronaut notices that a pendulum that took \(2.50 \mathrm{~s}\) for a complete cycle of swing when the rocket was waiting on the launch pad takes \(1.25 \mathrm{~s}\) for the same cycle of swing during liftoff. What is the acceleration of the rocket? (Hint: Inside the rocket, it appears that \(g\) has increased.)

A harmonic oscillator is made by using a \(0.600 \mathrm{~kg}\) frictionless block and an ideal spring of unknown force constant. The oscillator is found to have a period of \(0.150 \mathrm{~s}\) and a maximum speed of \(2 \mathrm{~m} / \mathrm{s} .\) Find \((\mathrm{a})\) the force constant of the spring and \((\mathrm{b})\) the amplitude of the oscillation.
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