Unit 5 - Problem 6 ==> Block Clock

Given:

• $$ \mbox{mass}  = 0.5  \mbox{kg} $$
• $$ k = 100 $$
• $$ \mbox{distance between springs} = 3  \mbox{meters} $$
• Switch  at midpoint and need to hit every second

Rationale:

• Ignore the width of the block
• Sequence: 1.5 m to spring, bounce on spring, then 1.5 m back to switch -- then again on opposite side
• $$ V_0 t = 2 * 1.5  \mbox{meters}  $$
• $$ t = \frac{3}{V_0}  $$
• Uses 0.5 of spring period
• $$ \frac{1}{2} T_{spring} = \frac{1}{2} (2 \pi) \sqrt\frac{m}{k}  $$
• $$ t +\frac{T}{2} = 1  \mbox{second} $$
• $$ \frac{3}{V_0} + \pi \sqrt\frac{m}{k} = 1  $$
• $$ \frac{3}{V_0}  = 1 - \pi \sqrt\frac{m}{k}  $$
• $$ V_0 = \frac{3}{( 1 - \pi \sqrt\frac{m}{k})} $$

Calculate:

• $$ V_0 =   \frac{3}{( 1 - \pi \sqrt\frac{0.5}{100})} $$
• $$ V_0 =   \frac{3}{( 1 - \pi * (.07071067811))} $$
• $$ V_0 =   \frac{3}{( 1 - 0.222 )} $$ 
•  $$ V_0 =   \frac{3}{.778} $$
• $$ V_0 = 3.8560  \mbox{meters/second} $$