Chapter 18, Problem 1P

To determine

Find the stiffness of each spring and the stiffest spring.

Answer to Problem 1P

The stiffness of each spring is $k_A=2.5\frac{\text{N}}{\text{mm}}$, $k_B=4\frac{\text{N}}{\text{mm}}$, and $k_C=1\frac{\text{N}}{\text{mm}}$, the stiffest spring is Spring B.

Explanation of Solution

Formula used:

Formula used to calculate the spring force is,

$F=kx$ (1)

Here,

$k$ is the stiffness factor or spring constant,

$x$ is the deflection.

Calculation:

Refer to figure Problem 18.1 in textbook, the graph with force-deflection relationships for three springs A, B and C are given.

Rearrange equation (1) to find $k$.

$k=\frac{F}{x}$ (2)

Refer to Figure Problem 18.1 in textbook,

For spring A,

At $x_A=4\text{mm, }{F}_A=10\text{N}$

For spring B,

At $x_B=10\text{mm, }{F}_B=40\text{N}$

For spring C,

At $x_C=10\text{mm, }{F}_C=10\text{N}$

Substitute $k_A$ for $k$, $10\text{N}$ for $F$, and $4\text{mm}$ for $x$ in equation (2),

$\begin{array}{c}{k}_A=\frac{10\text{N}}{4\text{mm}}\\ =2.5\frac{\text{N}}{\text{mm}}\end{array}$ (3)

Substitute $k_B$ for $k$, $40\text{N}$ for $F$, and $10\text{mm}$ for $x$ in equation (2),

$\begin{array}{c}{k}_B=\frac{40\text{N}}{10\text{mm}}\\ =4\frac{\text{N}}{\text{mm}}\end{array}$ (4)

Substitute $k_C$ for $k$, $10\text{N}$ for $F$, and $10\text{mm}$ for $x$ in equation (2),

$\begin{array}{c}{k}_C=\frac{10\text{N}}{10\text{mm}}\\ =1\frac{\text{N}}{\text{mm}}\end{array}$ (5)

Comparing the values in equation (3), (4) and (5), the spring B is the stiffest spring.

Therefore, the stiffness of each spring is $k_A=2.5\frac{\text{N}}{\text{mm}}$, $k_B=4\frac{\text{N}}{\text{mm}}$, and $k_C=1\frac{\text{N}}{\text{mm}}$, the stiffest spring is spring B.

Conclusion:

Thus, the stiffness of each spring is $k_A=2.5\frac{\text{N}}{\text{mm}}$, $k_B=4\frac{\text{N}}{\text{mm}}$, and $k_C=1\frac{\text{N}}{\text{mm}}$, the stiffest spring is spring B.