BIO Weighing Astronauts. This procedure has been used to “weigh” astronauts in space: A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

Solution 15E Step 1: The weight of the astronaut can be determined using the equation of time period. In the given problem, the spring has the spring constant k which is initially hanging with chair of mass m = 42.5 kg has its time period T =1.3 s. When the astronaut with mass m 1 1 2 sits on the chair, the time period change s to T =2.54 s, we need to calculate the mass of 2 the astronaut. Initially, Mass, m =142.5 kg Time period, T =113 s Final case, Mass , M=m +m 142.52kg+m 2 Time period, T =2.54 s 2 To find, Spring or force constant, k= Mass in final case, M= Mass of astronaut , m =2 Step 2: Time period is given by the equation, m T = 2 k ……………...1 Where T is time period, m is mass and k is force constant. 2 Solving for k, k = T m 2 On squaring both sides, k = 42m………….2 T Step 3: To calculate force constant using initial case, Substituting m =142.5 kg, T =113 s in equation 2, 42 k = (1.3 s).5 kg k = 991.79 N.m