# Chapter 4 - Exercises and Problems - Page 69: 62

The proofs are below.

#### Work Step by Step

a) We know that the force is: $F=k_tx$ In parallel, the spring constant is additive, so $k_t=k_1+k_2$. Thus: $F=(k_1+k_2)x$ b) We know that the force is: $F=k_tx$ In series, we can find $k_t$ to be: $\frac{1}{k_t}=\frac{1}{k_1}+\frac{1}{k_2}$ $\frac{1}{k_t}=\frac{k_1+k_2}{k_1k_2}$ $k_t=\frac{k_1k_2}{k_1+k_2}$ Thus, the force is: $F=(\frac{k_1k_2}{k_1+k_2})x$

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