Problem 8.29 Challenge

จาก \(ΣF_{r}= ma_{r}\)

\(-\frac{GMm}{r^{2}}= ma\)

\(a +\frac{GM}{r^{2}}= 0\)

\(a +\frac{G(\frac{4\pi r^{3}ρ}{3})}{r^{2}}= 0; ρ =\frac{M}{V}=\frac{3M}{4\pi r^{3}}\)

\(a + (\frac{4\pi ρG}{3})r = 0\)

\(a + \omega _{0}^{2}r = 0; \omega _{0} =\sqrt{\frac{4\pi ρG}{3}}\)

\(\therefore \omega _{0} =\sqrt{\frac{4\pi ρG}{3}}\)

\(\frac{2\pi}{T}=\sqrt{\frac{4\pi ρG}{3}}\)

\(T =\sqrt{\frac{3\pi}{ρG}}\;\blacksquare\)