85.689 kg is the astronaut's mass.
<u>Explanation:</u>
Given data: m = 22.5 kg and T = 1.3 sec
So, using the below formula,
![T = 2 \pi \sqrt{\frac{m}{k}}](https://tex.z-dn.net/?f=T%20%3D%202%20%5Cpi%20%5Csqrt%7B%5Cfrac%7Bm%7D%7Bk%7D%7D)
Now, after putting the values of m and T in the above equation, we will find out the value of k which is as,
![1.3 = 2 \times\left(\frac{22}{7}\right) \sqrt{\frac{22.5}{k}}](https://tex.z-dn.net/?f=1.3%20%3D%202%20%5Ctimes%5Cleft%28%5Cfrac%7B22%7D%7B7%7D%5Cright%29%20%5Csqrt%7B%5Cfrac%7B22.5%7D%7Bk%7D%7D)
![1.3 \times 7=44 \sqrt{\frac{22.5}{k}}](https://tex.z-dn.net/?f=1.3%20%5Ctimes%207%3D44%20%5Csqrt%7B%5Cfrac%7B22.5%7D%7Bk%7D%7D)
![\frac{9.1}{44}=\sqrt{\frac{22.5}{k}}](https://tex.z-dn.net/?f=%5Cfrac%7B9.1%7D%7B44%7D%3D%5Csqrt%7B%5Cfrac%7B22.5%7D%7Bk%7D%7D)
To remove the square root, take square on both sides, we get,
![(0.2068)^{2}=\frac{22.5}{k}](https://tex.z-dn.net/?f=%280.2068%29%5E%7B2%7D%3D%5Cfrac%7B22.5%7D%7Bk%7D)
![k=\frac{22.5}{0.0428}=525.7 \mathrm{N} / \mathrm{m}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B22.5%7D%7B0.0428%7D%3D525.7%20%5Cmathrm%7BN%7D%20%2F%20%5Cmathrm%7Bm%7D)
Now, we have the same string but this time we have different mass and different time. So, let the mass of the astronaut is
and
= 2.54 sec, k= 525.7 kg. Apply these values in the equation, we get,
![2.54 = 2 \times 3.14\left(\frac{m_{a}}{525.7}\right)^{\frac{1}{2}}](https://tex.z-dn.net/?f=2.54%20%3D%202%20%5Ctimes%203.14%5Cleft%28%5Cfrac%7Bm_%7Ba%7D%7D%7B525.7%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D)
![\frac{2.54}{6.28} = \left(\frac{m_{a}}{525.7}\right)^{\frac{1}{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B2.54%7D%7B6.28%7D%20%3D%20%5Cleft%28%5Cfrac%7Bm_%7Ba%7D%7D%7B525.7%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D)
![0.404 = \left(\frac{m_{a}}{525.7}\right)^{\frac{1}{2}}](https://tex.z-dn.net/?f=0.404%20%3D%20%5Cleft%28%5Cfrac%7Bm_%7Ba%7D%7D%7B525.7%7D%5Cright%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D)
Taking squares on both sides, we get,
![m_{a} = 525.7 \times(0.404)^{2}=525.7 \times 0.163 = 85.689 \mathrm{kg}](https://tex.z-dn.net/?f=m_%7Ba%7D%20%3D%20525.7%20%5Ctimes%280.404%29%5E%7B2%7D%3D525.7%20%5Ctimes%200.163%20%3D%2085.689%20%5Cmathrm%7Bkg%7D)