HumanPowered Centrifuge Space travel is fraught with hazards

Human-Powered Centrifuge Space travel is fraught with hazards, not the least of which are the many side effects of prolonged weightlessness, including weakened muscles, bone loss, decreased coordination, and unsteady balance. If you are fortunate enough to go on a trip to Mars, which could take more than a year each way, you might be a bit \"weak in the knees\" by the time you arrive. This could lead to problems when you try to take your first \"small step\" on the surface.

To counteract these effects, NASA is looking into ways to provide astronauts with \"portable gravity\" on long space flights. One method under consideration is the human-powered centrifuge, which not only subjects the astronauts to artificial gravity, but also gives them aerobic exercise. The device is basically a rotating, circular platform on which two astronauts lie supine along a diameter, head-to-head at the center, with their feet at opposite rims, as shown in the accompanying photo. The radius of the platform in this test model is 5.75 ft. As one astronaut pedals to rotate the platform, the astronaut facing the other direction can exercise in the artificial gravity. Alternatively, a third astronaut on a stationary bicycle can provide the rotation for the other two. While the astronauts\' feet are at the outer rim of the platform, their heads are near the center of the platform, and their hearts are 1.61 feet from the rim, which means that different parts of the astronauts\' bodies will experience different \"gravitational\" accelerations.

If the acceleration is 2.00 at the feet, what is the acceleration at the heart?

Solution

For circular motion,

the centripetal force, m*W^2/r = m*a

where W = angular speed of rotation

So, a = W^2*r

Now, at the feet,

a = 2*g = W^2*5.75

and at the heart let the acceleration be x

So, a = x = W^2*1.61

So, 2g/x = 5.75/1.61

So, x = 0.56*g <-----------option (d)