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:

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