Step 1

We are given the mass of crate is $m = 40\;{\rm{kg}}$ and spring constant is $k = 180\;{\rm{N/m}}$.

We are asked to determine the unstretched length of cable DB.

 
Step 2

The free-body of the given figure can be drawn as:

From the above figure, calculate the angle $\alpha$ and $\beta$ as:

\[\begin{array}{c} \tan \alpha = \left( {\frac{{2\;{\rm{m}}}}{{2\;{\rm{m}}}}} \right)\\ \alpha = {\tan ^{ - 1}}\left( 1 \right)\\ \alpha = 45^\circ \end{array}\]

Similarly,

\[\begin{array}{c} \tan \beta = \left( {\frac{{3\;{\rm{m}}}}{{2\;{\rm{m}}}}} \right)\\ \beta = {\tan ^{ - 1}}\left( {1.5} \right)\\ \beta = 56.3^\circ \end{array}\]
 
Step 3

Consider ${T_{CD}}$ and ${T_{BD}}$ are the forces acting on the member CD and BD as shown in the figure below,

To calculate the force in member BD, we need to apply the sine rule as:

\[\begin{array}{c} \frac{{mg}}{{\sin \left( {45^\circ + 56.3^\circ } \right)}} = \frac{{{T_{BD}}}}{{\sin \left( {90^\circ + 45^\circ } \right)}}\\ \frac{{\left( {40\;{\rm{kg}}} \right) \times \left( {9.81\;{\rm{m/}}{{\rm{s}}^2}} \right)}}{{\sin \left( {101.3^\circ } \right)}} = \frac{{{T_{BD}}}}{{\sin \left( {135^\circ } \right)}}\\ {T_{BD}} = \frac{{\left( {392.4\;{\rm{N}}} \right) \times \sin \left( {135^\circ } \right)}}{{\sin \left( {101.3^\circ } \right)}}\\ {T_{BD}} = 282.95\;{\rm{N}} \end{array}\]

To calculate the unstretched length of cable DB, we need to apply Hooke’s law as:

\[\begin{array}{c} {T_{BD}} = k\Delta x\\ {T_{BD}} = k\left( {{x_f} - {x_i}} \right) \end{array}\]

The stretched length (final length) can be calculated as:

\[\begin{array}{c} {x_f} = \sqrt {{{\left( {2\;{\rm{m}}} \right)}^2} + {{\left( {3\;{\rm{m}}} \right)}^2}} \\ {x_f} = \sqrt {13} \;{\rm{m}}\\ {x_f} = 3.60\;{\rm{m}} \end{array}\]

Substitute the values in the above expression, we get:

\[\begin{array}{c} \left( {282.95\;{\rm{N}}} \right) = \left( {180\;{\rm{N/m}}} \right) \times \left( {3.60\;{\rm{m}} - {x_i}} \right)\\ 1.57\;{\rm{m}} = \left( {3.60\;{\rm{m}} - {x_i}} \right)\\ {x_i} = 2.03\;{\rm{m}} \end{array}\]