Use Charles Law: V1/T1 = V2/T2
0.30 m^3/27 C = V2/127 C
27V2 = 127 * 0.3
V2= 38.1/27 = 1.4 m^3
Solution :
Acceleration due to gravity of the earth, g ![$=\frac{GM}{R^2}$](https://tex.z-dn.net/?f=%24%3D%5Cfrac%7BGM%7D%7BR%5E2%7D%24)
![$g=\frac{G(4/3 \pi R^2 \rho)}{R^2}=G(4/3 \pi R \rho)$](https://tex.z-dn.net/?f=%24g%3D%5Cfrac%7BG%284%2F3%20%5Cpi%20R%5E2%20%5Crho%29%7D%7BR%5E2%7D%3DG%284%2F3%20%5Cpi%20R%20%5Crho%29%24)
Acceleration due to gravity at 1000 km depths is :
![$g=G\left(\frac{4}{3}\pi (R-d) \rho\right)$](https://tex.z-dn.net/?f=%24g%3DG%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Cpi%20%28R-d%29%20%5Crho%5Cright%29%24)
![$g=6.67 \times 10^{-11}\left(\frac{4}{3}\times 3.14 \times (6371-1000) \times 5.5 \times 10^3\right)$](https://tex.z-dn.net/?f=%24g%3D6.67%20%5Ctimes%2010%5E%7B-11%7D%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Ctimes%203.14%20%5Ctimes%20%286371-1000%29%20%5Ctimes%205.5%20%5Ctimes%2010%5E3%5Cright%29%24)
![$= 822486 \times 10^{-8}$](https://tex.z-dn.net/?f=%24%3D%20822486%20%5Ctimes%2010%5E%7B-8%7D%24)
![$=0.822 \times 10^{-2} \ km/s$](https://tex.z-dn.net/?f=%24%3D0.822%20%5Ctimes%2010%5E%7B-2%7D%20%5C%20km%2Fs%24)
= 8.23 m/s
Acceleration due to gravity at 2000 km depths is :
![$g=G\left(\frac{4}{3}\pi (R-d) \rho\right)$](https://tex.z-dn.net/?f=%24g%3DG%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Cpi%20%28R-d%29%20%5Crho%5Cright%29%24)
![$g=6.67 \times 10^{-11}\left(\frac{4}{3}\times 3.14 \times (6371-2000) \times 5.5 \times 10^3\right)$](https://tex.z-dn.net/?f=%24g%3D6.67%20%5Ctimes%2010%5E%7B-11%7D%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Ctimes%203.14%20%5Ctimes%20%286371-2000%29%20%5Ctimes%205.5%20%5Ctimes%2010%5E3%5Cright%29%24)
![$= 673552 \times 10^{-8}$](https://tex.z-dn.net/?f=%24%3D%20673552%20%5Ctimes%2010%5E%7B-8%7D%24)
![$=0.673 \times 10^{-2} \ km/s$](https://tex.z-dn.net/?f=%24%3D0.673%20%5Ctimes%2010%5E%7B-2%7D%20%5C%20km%2Fs%24)
= 6.73 m/s
Acceleration due to gravity at 3000 km depths is :
![$g=G\left(\frac{4}{3}\pi (R-d) \rho\right)$](https://tex.z-dn.net/?f=%24g%3DG%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Cpi%20%28R-d%29%20%5Crho%5Cright%29%24)
![$g=6.67 \times 10^{-11}\left(\frac{4}{3}\times 3.14 \times (6371-3000) \times 5.5 \times 10^3\right)$](https://tex.z-dn.net/?f=%24g%3D6.67%20%5Ctimes%2010%5E%7B-11%7D%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Ctimes%203.14%20%5Ctimes%20%286371-3000%29%20%5Ctimes%205.5%20%5Ctimes%2010%5E3%5Cright%29%24)
![$= 3371 \times 153.86 \times 10^{-8}$](https://tex.z-dn.net/?f=%24%3D%203371%20%5Ctimes%20153.86%20%5Ctimes%2010%5E%7B-8%7D%24)
= 5.18 m/s
Acceleration due to gravity at 4000 km depths is :
![$g=G\left(\frac{4}{3}\pi (R-d) \rho\right)$](https://tex.z-dn.net/?f=%24g%3DG%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Cpi%20%28R-d%29%20%5Crho%5Cright%29%24)
![$g=6.67 \times 10^{-11}\left(\frac{4}{3}\times 3.14 \times (6371-4000) \times 5.5 \times 10^3\right)$](https://tex.z-dn.net/?f=%24g%3D6.67%20%5Ctimes%2010%5E%7B-11%7D%5Cleft%28%5Cfrac%7B4%7D%7B3%7D%5Ctimes%203.14%20%5Ctimes%20%286371-4000%29%20%5Ctimes%205.5%20%5Ctimes%2010%5E3%5Cright%29%24)
![$= 153.84 \times 2371 \times 10^{-8}$](https://tex.z-dn.net/?f=%24%3D%20153.84%20%5Ctimes%202371%20%5Ctimes%2010%5E%7B-8%7D%24)
![$=0.364 \times 10^{-2} \ km/s$](https://tex.z-dn.net/?f=%24%3D0.364%20%5Ctimes%2010%5E%7B-2%7D%20%5C%20km%2Fs%24)
= 3.64 m/s