Unit 5 - Problem 5 ==> Falling Thru The Earth

Given:

• $$ F_e = g m \frac{r}{r_e}  $$
• $$ g = 10 $$
• $$ r_e = 6,400  \mbox{km} = 6,400,000  \mbox{meters} $$

Rationale:

• $$ F_{restoring} = k x = r \frac{g m}{r_e} $$
• $$ k = \frac{g m}{r_e} $$
• $$ \frac{1}{k} = \frac{r_e}{g m} $$
• $$ x = r $$ 
• $$ T_{period} = 2 \pi \sqrt{(m \frac{1}{k})} $$
• $$ T_{period} = 2 \pi \sqrt{(m \frac{r_e}{g m})} $$
• $$ T_{period} = 2 \pi \sqrt\frac{r_e}{g} $$

Calculate:

• $$ T_{period} = 2 \pi \sqrt\frac{r_e}{g} $$
• $$ T_{period} = \frac{44}{7} \sqrt\frac{6,400,000}{10} $$
• $$ T_{period} = 6.2857143 \sqrt{640,000} $$
• $$ T_{period} = 6.2857143 * 800 = 5,028.57  \mbox{seconds}$$
• $$ T_{period} =83.8  \mbox{minutes} $$