Problem 120 - Precalculus

Question

Refer to the following: NASA explores artificial gravity as a way to counter the physiologic effects of extended weightlessness for future space exploration. NASA's centrifuge has a 58 -foot-diameter arm. If two humans are on opposite (red and blue) ends of the centrifuge and they rotate one full rotation every second, what is their linear speed in feet per second?

Step-by-Step Solution

Answer

The linear speed is approximately 182.12 feet per second.

Step 1: Determine the Radius of the Centrifuge

Given that the diameter of the centrifuge is 58 feet, the radius is half of the diameter. To find the radius, we divide the diameter by 2, thus:\[\text{Radius} = \frac{58}{2} = 29 \text{ feet}\]

Step 2: Calculate the Circumference of the Circle

The circumference of a circle is calculated using the formula:\[\text{Circumference} = 2\pi r\]Substituting the value of the radius:\[\text{Circumference} = 2\pi \times 29 = 58\pi \text{ feet}\]

Step 3: Determine the Linear Speed

The linear speed is defined as the distance traveled per unit of time. If the centrifuge completes one full rotation every second, then the linear speed is the same as the circumference of the circle per second. Thus the linear speed is:\[\text{Linear Speed} = 58\pi \text{ feet per second}\]

Step 4: Calculate the Approximate Linear Speed

To find the approximate numerical value of the linear speed, we can use the approximation \(\pi \approx 3.14\). Therefore:\[\text{Linear Speed} = 58 \times 3.14 \approx 182.12 \text{ feet per second}\]

Key Concepts

Radius of a Circle

The radius of a circle is one of its most fundamental properties. It is defined as the distance from the center of the circle to any point on its perimeter. Understanding this concept is crucial for many geometrical calculations.

For example, if you are given the diameter, which is the length across the circle through its center, you can easily find the radius by dividing the diameter by two. This is because the diameter is twice the radius.

In the context of the NASA centrifuge, with a diameter of 58 feet, the radius is:

• Divide the diameter by 2:
  1. Radius = Diameter / 2
  2. Hence, Radius = 58 / 2 = 29 feet

Having the correct radius allows us to compute other important circle properties, such as the circumference and area, which are beneficial for various practical applications, from engineering to space exploration.

Circumference of a Circle

The circumference is the total distance around the circle, much like the perimeter is for polygons. It tells you how far you would travel if you walked along the edge of the circle. This measure is vital for understanding how much space an object, like NASA's centrifuge, takes up along its circular path.

To calculate the circumference, you use the formula:

• 1. Circumference \( = 2\pi r \)
  2. Where \( \pi \approx 3.14 \) is a constant that represents the ratio of the circumference of any circle to its diameter.

For a circle with a radius of 29 feet, such as the centrifuge:

• Circumference = \( 2 \times \pi \times 29 = 58\pi \) feet
• This means if the centrifuge arm made one complete revolution, it would cover 58\pi feet or approximately 182.12 feet, given \( \pi \approx 3.14 \).

This calculation of the circumference is critical for determining linear speed when the circular motion is involved.

Centripetal Force

Centripetal force is the force required to keep an object moving in a circular path. It's "centripetal" because it points towards the center of the circle. When astronauts use a centrifuge for training or research, they experience this force, which helps prepare them for the intense gravitational forces encountered in space.

The magnitude of centripetal force depends on:

• The object's mass
• The speed at which the object is moving
• The radius of the path

The formula to calculate this force is:

• Centripetal Force \( F = \frac{mv^2}{r} \)

Where:

• \( m \) is the mass of the object
• \( v \) is its linear speed
• \( r \) is the radius of the circular path

In the case of the centrifuge, to remain on the circular path at high speeds, they rely on this inward force. Understanding how centripetal force works in conjunction with other factors like radius and linear speed is essential in designing safe and effective space training programs.