Problem 119 - 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 their linear speed is 200 miles per hour, how fast is the arm rotating? Express the answer in radians per second to two significant digits.

Step-by-Step Solution

Answer

The arm is rotating at approximately 10 radians per second.

Step 1: Convert Diameter to Radius

The diameter of the centrifuge's arm is given as 58 feet. To find the radius, divide the diameter by 2: \[ r = \frac{58}{2} = 29 \text{ feet} \].

Step 2: Convert Linear Speed to Feet per Second

The linear speed is given as 200 miles per hour (mph). First, convert this speed into feet per second. Since 1 mile = 5280 feet and 1 hour = 3600 seconds, use the conversion:\[ 200 \times \frac{5280}{3600} = \frac{200 \times 5280}{3600} = \frac{1056000}{3600} \approx 293.33 \text{ feet per second} \].

Step 3: Use Linear Speed to Find Angular Speed

The formula to relate linear speed \( v \), radius \( r \), and angular speed \( \omega \) is \( v = \omega \times r \). Solve for \( \omega \):\[ \omega = \frac{v}{r} = \frac{293.33}{29} \approx 10.12 \text{ radians per second} \].

Step 4: Round Answer to Two Significant Digits

The calculated angular speed is \( 10.12 \text{ radians per second} \). Round this to two significant digits to obtain \( 10 \text{ radians per second} \).

Key Concepts

Linear Speed

Linear speed refers to how fast an object is moving along a straight path. In the context of a centrifuge, this is the speed at which the astronauts would be moving while attached to the rotating arm. Linear speed is typically measured in units like miles per hour (mph) or feet per second (ft/s). In this exercise, the linear speed is initially given in miles per hour, which we need to convert into feet per second in order to relate it to the angular speed. To do this conversion:

• 1 mile equals 5280 feet
• 1 hour equals 3600 seconds

By using the conversion of \(200 \text{ mph} \times \frac{5280}{3600} \approx 293.33 \text{ ft/s}\),we find the linear speed in a unit that matches the radius's unit (feet), allowing for the calculation of angular speed.

Radius Conversion

The radius is a crucial part of calculating angular speed, as it serves as the pivot distance from the center of rotation to the moving edge. Often, unlike diameter, the measurements in physics problems require conversion to radius to use physical formulas effectively. The original problem provides the diameter of the centrifuge's arm. The diameter spans across the entire circle, while the radius is half the diameter.To find the radius, you simply divide the diameter by 2, applying the formula:\[r = \frac{\text{diameter}}{2}\].Given a diameter of 58 feet, the radius becomes:\[r = \frac{58}{2} = 29 \text{ feet}\].With the correct radius, other calculations such as angular speed become simple and precise.

Radians per Second

Radians per second is a unit of angular speed, showing how quickly an object is rotating. This measurement gives the angle an object sweeps through, measured in radians, per unit of time (seconds). Unlike degrees, radians are a more natural way of expressing angles since they are directly related to the properties of circles.To find how fast the centrifuge arm is rotating in terms of radians per second, we use the relationship:\(v = \omega \times r\),where \( v \) is linear speed, \( \omega \) is angular speed, and \( r \) is radius. Rearranging gives:\(\omega = \frac{v}{r}\).Plugging in the known values:\[\omega = \frac{293.33 \text{ ft/s}}{29 \text{ ft}} \approx 10.12 \text{ radians per second}\].When rounded to two significant digits, it simplifies to 10 radians per second. This measurement, 10 radians per second, tells us how swiftly the centrifuge arm is spinning.