Determine the stretch in each of the two springs required to hold the 20-kg crate in the equilibrium position shown. Each spring has an unstretched length of 2 m and a stiffness of k = 300 N/m.

#### Solution:

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We will first write each force in the cables/springs in Cartesian vector form. To do so, we need to figure out the position vectors for each force.

Notice in the diagram that springs OA and OB lie in the x and y axis, respectively. Therefore, we only need to find a position vector for cable OC.

r_{OC}\,=\,\left\{6i+4j+12k\right\} m

Now, we calculate the magnitude:

magnitude of r_{OC}\,=\,\sqrt{(6)^2+(4)^2+(12)^2}\,=\,14 m

Now, we write the unit vector for the position vector.

u_{AB}\,=\,\left(\dfrac{6}{14}i+\dfrac{4}{14}j+\dfrac{12}{14}k\right)

We can now express the force in the cable OC in Cartesian form:

F_{OC}\,=\,\left\{0.428F_{OC}i+0.286F_{OC}j+0.857F_{OC}k\right\}

Now the other forces:

F_{OA}\,=\,\left\{0i-F_{OA}j+ok\right\}

F_{OB}\,=\,\left\{-F_{OB}i+0j+ok\right\}

F\,=\,\left\{0i+0j-(20)(9.81)k\right\}\,=\,\left\{0i+0j-196.2k\right\}

(Force F, is the force exerted on the ring O by the crate, which has a mass of 20 kg. It is applied directly downwards, meaning it only has a z-component)

Now that we have written all forces in Cartesian vector form, we can write our equation of equilibrium. All forces added together must equal zero because the system is in equilibrium.

F_{OC}+F_{OA}+F_{OB}+F\,=\,0

As all the forces added together equals zero, then each component (x,y, z-components) added individually must also equal zero.

0.429F_{OC}-F_{OB}\,=\,0

y-components:

0.286F_{OC}-F_{OA}\,=\,0

z-components:

Solving the three equations gives us:

F_{OB}\,=\,98 N

F_{OA}\,=\,65.5 N

We can now find the stretch of the springs using the following equation (Hooke’s Law):

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

Starting with the force in spring OA:

65.5\,=\,300s

(solve for s)

s\,=\,0.218 m

Now, the force in spring OB:

98\,=\,300s

(solve for s)

s\,=\,0.327 m

#### Final Answers:

Stretch of spring OB = 0.327 m