Chapter 8.1, Problem 13P

To determine

To find the natural length of the spring.

Answer to Problem 13P

The natural length of the spring is $\text{13 cm}$ .

Explanation of Solution

Given information:

The given statement is:

“A load of $\text{12 kg}$ stretches a spring to a total length of $\text{15 cm}$ , and a load of $\text{30 kg}$ stretches it to a length of $\text{18 cm}$ . Find the natural length of the spring.”

Calculation :

Let the natural length of the spring be $l$ .

  $\begin{array}{l}\frac{\text{Force}\left(F\right)}{\text{Elongation}\left(x\right)}=k\left[F=mg\right]\\ \text{or},mg=kx\end{array}$

The natural length of the spring can be calculated by:

  $\begin{array}{l}\frac{m_1}{x_1}=\frac{m_2}{x_2}\\ \frac{12}{15-l}=\frac{30}{18-l}\\ 12\left(18-l\right)=30\left(15-l\right)\\ 216-12l=450-30l\\ 30l-12l=450-216\\ 18l=234\\ l=\frac{234}{18}\\ l=13\end{array}$

Hence, the natural length of the spring is $\text{13 cm}$ .