Problem 38 - 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 897.3 N/m, and the astronaut's mass is approximately 79.1 kg.

Step 1: 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.

Step 2: 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:\[k = \frac{4\pi^2 \times 35.4}{(1.25)^2} \approx 897.3 \text{ N/m}\]

Step 3: 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 \( m_a \):\[\frac{(2.23)^2}{4\pi^2} \times 897.3 = 35.4 + m_a\]\[m_a = \frac{(2.23)^2}{4\pi^2} \times 897.3 - 35.4\]\[m_a \approx 79.1 \text{ kg}\]

Key Concepts

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’s 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’s gravitational pull.