# The spring has a stiffness of k = 800 N/m

The spring has a stiffness of k = 800 N/m and an unstretched length of 200 mm. Determine the force in cables BC and BD when the spring is held in the position shown.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

#### Solution:

We will first calculate the force of the spring.

From the diagram, we can see that the length of the spring is 500 mm. We also know that the unstretched length of the spring is 200 mm. Therefore, the spring is currently stretched by 300 mm (500 – 200 = 300 mm).

The force of the spring can be calculated using Hook’s Law.

$F\,=\,ks$

(Where $F$ is force, $k$ is the stiffness of the spring, and $s$ is the stretch of the spring)

$F\,=\,(800)(0.3)$

(Note that we converted 300 mm to 0.3 m)

$F\,=\,240$ N

Let us now draw our free body diagram.

The angles were calculated using trigonometry, specifically, the inverse of tan (arctan).

The brown angle was found by:

$\text{tan}^{-1}\left(\dfrac{400}{400}\right)=45^0$

The orange angle was found by:

$\text{tan}^{-1}\left(\dfrac{300}{400}\right)=36.87^0$

We can now write our equations of equilibrium. We will assume forces going $\rightarrow^+$ to be positive and $\uparrow+$ to be positive.

$\rightarrow ^+\sum \text{F}_\text{x}\,=\,0$

$F_{BC}\text{cos}\,(45^0)\,+\,F_{BD}\text{cos}\,(36.87^0)\,-\,240\,=\,0$ (eq.1)

$+\uparrow \sum \text{F}_\text{y}\,=\,0$

$F_{BC}\text{sin}\,(45^0)\,-\,F_{BD}\text{sin}\,(36.87^0)\,=\,0$ (eq.2)

Show me the free body diagram

Let us solve for $F_{BC}$ and $F_{BD}$.

Isolate for $F_{BC}$ in eq.2.

$F_{BC}\,=\,\dfrac{F_{BD}\text{sin}\,(36.87^0)}{\text{sin}\,(45^0)}$ (eq.3)

Substitute this value into eq.1.

$\dfrac{F_{BD}\text{sin}\,(36.87^0)}{\text{sin}\,(45^0)}\text{cos}\,(45^0)\,+\,F_{BD}\text{cos}\,(36.87^0)\,-\,240\,=\,0$

(solve for $F_{BD}$)

$F_{BD}\,=\,171.4$ N

Substitute the value of $F_{BD}$ we just found into eq.3 to find $F_{BC}$

$F_{BC}\,=\,\dfrac{171.4\text{sin}\,(36.87^0)}{\text{sin}\,(45^0)}$

$F_{BC}\,=\,145.4$ N