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:

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