In space, astronauts need to work out about two hours a day to combat the muscle and bone density wasting effects of zero gravity. With no gravity, how are you supposed to workout? Don’t you need gravity to hold you in place to do a workout? Until the technology for artificial gravity is invented, NASA had to get a little creative to help astronauts workout. A harness with resistance bands attached to either side (pictured below) is used to hold astronauts in place to run on a treadmill. To most effectively reduce bone density loss, the resistance bands must produce enough downward force to simulate the gravity on Earth, a 1G environment. To choose resistance bands that are suitable for this purpose, you must know the total force that they need to withstand. So, how do you calculate the total tension in each band so that there is a 1G environment?

Assuming that the astronaut is of average height and mass (H=1.77 meters, m=80 kg), we can determine the downward force the required for the 1G environment and how high the attachment point is (hips). According to the anthropometric data on the document attached (see bottom for links), we know that the height of the hips are 0.53 of his total height, equaling 0.9381 meters.

To determine what a 1G environment is we use F=ma (where a is the acceleration due to earth’s gravity, 9.8 m/s^2). So now we know that the force in the y direction must equal 784 N. Since there are two bands, each band is only responsible for producing half of that downward force, 392 N.

Before we solve for the tension in the band, we need to know the angle theta. Using simple trigonometry, theta = sin^-1(.9381) = 69.73 degrees. You can also use the width of the treadmill (.7 meters) with the height to find the hypotenuse and then use the Law of Sines to find theta. Either way will get you the same answer.

Now we have everything we need to solve for T.

T_y = T * Sin(theta)

T_y = 392 N (downforce), 392 = T * Sin(69.73)

T = 417.9 N.

This means that each resistance band must withstand 417.9 N of force to produce a 1G environment for a man who is 1.77 meters tall and 80 kg.

Assuming average height and weight is good because most astronauts are very close to the average height and weight of US males. There is not a lot of storage space on the international space station so multiple harnesses is not really plausible. By choosing resistance bands for average height and weight will be able to accommodate most astronauts adequately. I also simplified the body to a point mass, this changes the the angle of theta a little. In this application, I think this simplification is okay because it wouldn’t change the overall tension by a by much. A safety factor would easily cover the change in force that the angle is responsible for.

This information can now be used as a metric to select the proper material and dimensions for the resistance bands in the harness.

Below is a chart of the anthropometric data mentioned above, and my hand-written calculations for this problem.

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