Chapter 6, Problem 125P

(a)

To determine

The effective spring constant of the series combination.

Answer to Problem 125P

The effective spring constant of the series combination is $\frac{k}{2}$.

Explanation of Solution

Assume that the two springs have springs constants $k_1$ and $k_2$ respectively. The force that each spring experiences will be the same (otherwise the springs will buckle) and it will be the same as the force on the mass.

Write the equation for the spring force of the spring with spring constant $k_1$.

  $F=-k_1{x}_1$

Here, $F$ is the spring force and $x_1$ is the compression of the spring with spring constant $k_1$.

Rewrite the above equation for $x_1$ .

  $x_1=-\frac{F}{k_1}$        (I)

Write the equation for the spring force of the spring with spring constant $k_2$ .

  $F=-k_2{x}_2$

Here, $x_2$ is the compression of the spring with spring constant $k_2$ .

Rewrite the above equation for $x_2$ .

  $x_2=-\frac{F}{k_2}$        (II)

Write the equation for the force on the mass.

  $F=-k_{\text{eq}}x$

Here, $k_{\text{eq}}$ is the effective spring constant of the series combination and $x$ is the net displacement.

Rewrite the above equation for $x$ .

  $x=-\frac{F}{k_{\text{eq}}}$        (III)

Write the equation for the net displacement.

  $x=x_1+x_2$

Put equations (I) and (II) in the above equation.

  $\begin{array}{c}x=-\frac{F}{k_1}+\left(-\frac{F}{k_2}\right)\\ =-\frac{F}{k_1}-\frac{F}{k_2}\end{array}$

Put the above equation in equation (III).

  $\begin{array}{c}-\frac{F}{k_1}-\frac{F}{k_2}=-\frac{F}{k_{\text{eq}}}\\ \frac{1}{k_{\text{eq}}}=\frac{1}{k_1}+\frac{1}{k_2}\end{array}$        (IV)

Conclusion:

The spring constants of the two springs are equal and is equal to $k$ .

Substitute $k$ for $k_1$ and $k_2$ in equation (IV) to find $k_{\text{eq}}$ .

  $\begin{array}{c}\frac{1}{k_{\text{eq}}}=\frac{1}{k}+\frac{1}{k}\\ =\frac{2k}{k^2}\\ k_{\text{eq}}=\frac{k}{2}\end{array}$

Therefore, the effective spring constant of the series combination is $\frac{k}{2}$ .

(b)

To determine

The effective spring constant of the parallel combination.

Answer to Problem 125P

The effective spring constant of the parallel combination is $2k$ .

Explanation of Solution

In parallel combination, the force on the mass will be equal to the sum of the two individual spring forces.

Write the expression for the force on the mass.

  $F=F_1+F_2$        (V)

Here, $F$ is the force on the mass, $F_1$ is the spring force of the spring with spring constant $k_1$ and $F_2$ is the spring force of the spring with spring constant $k_2$ .

The two springs will be compressed along the same distance in parallel combination.

Write the equation for the spring force of the spring with spring constant $k_1$ .

  $F_1=-k_{1}x$

Here, $x$ is the amount of compression

Write the equation for the spring force of the spring with spring constant $k_2$ .

  $F_2=-k_{2}x$

Write the equation for the force on the mass.

  $F=-k_{\text{eq}}x$

Put the above three equations in equation (V).

  $\begin{array}{c}-k_{\text{eq}}x=-k_{1}x+\left(-k_{2}x\right)\\ =-k_{1}x-k_{2}x\\ k_{\text{eq}}=k_1+k_2\end{array}$        (VI)

Conclusion:

Substitute $k$ for $k_1$ and $k_2$ in equation (VI) to find $k_{\text{eq}}$ .

  $\begin{array}{c}{k}_{\text{eq}}=k+k\\ =2k\end{array}$

Therefore, the effective spring constant of the parallel combination is $2k$.