Chapter 3.3, Problem 17P

Determine the stiffness k_T of the single spring such that the force F will stretch it by the same amount s as the force F stretches the two springs. Express k_T in terms of stiffness k₁, and k₂ of the two springs.

To determine

The stiffness $k_T$ of the single spring.

Answer to Problem 17P

The stiffness $k_T$ of the single spring, $\underset{\_}{\frac{1}{k_T}=\frac{1}{k_1}+\frac{1}{k_2}}$.

Explanation of Solution

Given information:

• The force F will stretch it by the same amount s as the force F stretches the two springs.

Write the magnitude of force exerted on a linearly elastic spring in an unstretched position.

$\begin{array}{l}F=ks\\ s=\frac{F}{k_T}\end{array}$

Here, the stiffness of the spring is k and the stretch of the spring is s.

Write the stretch value for two springs.

$s=s_1+s_2$ (I)

Here, the stretch of the first spring is $s_1$ and the stretch of the first spring is $s_2$.

Write the equation for the magnitude of force exerted on a linearly elastic spring 1.

$\begin{array}{l}F=k_1{s}_1\\ s_1=\frac{F}{k_1}\end{array}$

Here, the stiffness of the spring 1 is $k_1$.

Write the equation for the magnitude of force exerted on a linearly elastic spring 2.

$\begin{array}{l}F=k_2{s}_2\\ s_2=\frac{F}{k_2}\end{array}$

Here, the stiffness of the spring 2 is $k_2$.

Conclusion:

Substitute $\frac{F}{k_T}$ for s, $\frac{F}{k_1}$ for $s_1$, and $\frac{F}{k_2}$ for $s_2$ in Equation (I).

$\begin{array}{l}\frac{F}{k_T}=\frac{F}{k_1}+\frac{F}{k_2}\\ F\left(\frac{1}{k_T}\right)=F\left(\frac{1}{k_1}+\frac{1}{k_2}\right)\\ \frac{1}{k_T}=\frac{1}{k_1}+\frac{1}{k_2}\end{array}$

Thus, the stiffness $k_T$ of the single spring is $\underset{\_}{\frac{1}{k_T}=\frac{1}{k_1}+\frac{1}{k_2}}$.