Unit 5 - Problem 8 ==> Springs In Series

Given:

• Two springs in series with spring constants k-1 and k-2.
• mass of m attached to spring 2.

Rationale:

• The k of the combined springs must be less than the k of either spring
• $$ F_g = m g    $$
• $$ F_{spring_2}  = m g $$
• $$ \Delta{x_{total}} = \Delta{x_2} + \Delta{x_1} $$
• $$\mbox{In general:  } \Delta{x} = \frac{F}{k} $$
• $$ \frac{m g}{k_{total}} = \frac{m g}{k_2} + \frac{m g}{k_1} $$
• $$ \frac{1}{k_{total}} =  \frac{1}{k_2} + \frac{1}{k_1} $$
• $$ \frac{1}{k_{total}} =  \frac{(k_2 + k_1)}{(k_1 * k_2)}  $$
• $$ k_{total} = \frac{(k_1 * k_2)}{(k_2 + k_1)} $$

Testcase: k-1 = k-2

• $$ k_1 = k_2 $$
• $$ k_{total}  = \frac{k_1^2}{2 k_1} $$
• $$ k_{total} = \frac{k_1}{2} $$

Testcase: k-1 <<  k2

• $$ k_1 = 100 $$
• $$ k_2 =  100,000 $$
• $$ k_{total} = \frac{(k_1 * k_2)}{(k_2 + k_1)} $$
•  $$ k_{total} = \frac{(100 * 100000)}{(100000 + 100)} $$
•  $$ k_{total} = 99.9 $$

Other References

• From Wolfram