The 20-g centrifuge at NASA's Ames Research Center in Mountain View, California, is a cylindrical tube 58 ft long with a radius of 29 ft (see figure). If a rider sits in a chair at the end of one arm facing the center, how many revolutions per minute would be required to create a horizontal normal force equal in magnitude to 20.0 times the rider's weight?
-rev/min
To determine the number of revolutions per minute (rev/min) required to create a horizontal normal force equal to 20.0 times the rider's weight in the 20-g centrifuge at NASA's Ames Research Center, we can use the following steps:
1. Convert the acceleration due to gravity into the equivalent acceleration in terms of "g." The acceleration due to gravity is approximately 9.8 m/s² or 32.2 ft/s². Since this is a 20-g centrifuge, we multiply the acceleration due to gravity by 20: 20 * 32.2 ft/s² = 644 ft/s².
2. Determine the magnitude of the centrifugal force acting on the rider. The centrifugal force (Fc) is given by the equation: Fc = m * a, where "m" is the rider's mass and "a" is the acceleration. In this case, we want Fc to be equal to 20 times the rider's weight (20 * W). Therefore, Fc = 20 * W.
3. Calculate the rider's weight. The weight (W) of an object is given by the equation: W = m * g, where "m" is the mass and "g" is the acceleration due to gravity. Since we know the weight is equal to m * g, we can rearrange the equation to solve for "m": m = W / g.
4. Substitute the expression for mass in the equation for centrifugal force: Fc = 20 * (W / g) * a.
5. Now, we can substitute the expression for acceleration from step 1 into the equation for centrifugal force: Fc = 20 * (W / g) * (644 ft/s²).
6. Rearrange the equation to solve for Fc: Fc = (20 * W * 644 ft/s²) / g.
7. Since the centrifugal force is equal to the horizontal normal force, we can substitute the expression for centrifugal force into the equation: Fc = m * (v² / r), where "m" is the rider's mass, "v" is the velocity, and "r" is the radius of the centrifuge.
8. Rearrange the equation to solve for the velocity: v = sqrt((Fc * r) / m).
9. Substitute the expression for Fc from step 6 and the given radius into the equation: v = sqrt((((20 * W * 644 ft/s²) / g) * 29 ft) / m).
10. Finally, since we're looking for the number of revolutions per minute, we can convert the velocity into the circumference of the ride and divide by the time: rev/min = (2 * pi * r * v) / (2 * pi * L / t), where "L" is the length of the centrifuge and "t" is the time taken for one revolution.
To solve this equation, we need the time taken for one revolution. Unfortunately, the time information is not provided in the question. Without the time value, we cannot calculate the exact number of revolutions per minute.