Question

use what you know about newtons law of universal to predict the values for his missing variables.
system
kepler - 92
gliese - 597
kepler - 731
fg
2.7×10²⁶n
8.03×10²⁴n
choose
m₁
2.3×10²⁷kg
choose
4.17×10²⁵kg
m₂
choose
5.6×10³⁰kg
6.4×10³⁵kg
r
5.7×10¹⁰m
1.93×10¹¹m
1.17×10¹¹m

Explanation:

Step1: Recall Newton's law of universal gravitation

Newton's law of universal gravitation is given by the formula \(F_{g}=G\frac{m_{1}m_{2}}{r^{2}}\), where \(G = 6.67\times10^{- 11}\space N\cdot m^{2}/kg^{2}\)

Step2: Solve for \(m_{2}\) in the Kepler - 92 system

We are given \(F_{g}=2.7\times 10^{26}\space N\), \(m_{1}=2.3\times 10^{27}\space kg\), \(r = 5.7\times 10^{10}\space m\) and \(G=6.67\times 10^{-11}\space N\cdot m^{2}/kg^{2}\)

From \(F_{g}=G\frac{m_{1}m_{2}}{r^{2}}\), we can solve for \(m_{2}\) as \(m_{2}=\frac{F_{g}r^{2}}{Gm_{1}}\)

Substitute the values:

$$
LATEXBLOCK0
$$

Step3: Solve for \(m_{1}\) in the Gliese - 597 system

We are given \(F_{g}=8.03\times 10^{24}\space N\), \(m_{2}=5.6\times 10^{30}\space kg\), \(r = 1.93\times 10^{11}\space m\) and \(G = 6.67\times 10^{-11}\space N\cdot m^{2}/kg^{2}\)

From \(F_{g}=G\frac{m_{1}m_{2}}{r^{2}}\), we can solve for \(m_{1}\) as \(m_{1}=\frac{F_{g}r^{2}}{Gm_{2}}\)

Substitute the values:

$$
LATEXBLOCK1
$$

Step4: Solve for \(F_{g}\) in the Kepler - 731 system

We are given \(m_{1}=4.17\times 10^{25}\space kg\), \(m_{2}=6.4\times 10^{35}\space kg\), \(r = 1.17\times 10^{11}\space m\) and \(G = 6.67\times 10^{-11}\space N\cdot m^{2}/kg^{2}\)

From \(F_{g}=G\frac{m_{1}m_{2}}{r^{2}}\)

Substitute the values:

$$
LATEXBLOCK2
$$

Answer:

• For Kepler - 92: \(m_{2}\approx5.6\times 10^{30}\space kg\)
• For Gliese - 597: \(m_{1}\approx8.0\times 10^{26}\space kg\)
• For Kepler - 731: \(F_{g}\approx1.3\times 10^{29}\space N\)