Determine the mass of each of the two cylinders if they cause a sag of s = 0.5 m when suspended from the rings at A and B. Note that s = 0 when the cylinders are removed.

#### Solution:

We will first find the length of the **unstretched** spring. To do so, we will use the Pythagorean theorem.

spring length = 2.5 m

let us now find the stretch of the spring after the weights were attached.

(Remember that with the weights attached, s=0.5m, which means the vertical component is now 2 m)

spring length = 2.83 m

Therefore, we now know that when the weights were attached, the spring stretched by 2.83 m – 2.5 m = 0.33 m.

With this result, we can figure out the force of the spring, using Hook’s Law. Hook’s Law states:

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

F\,=\,(100)(0.33)F\,=\,33 N

We can now draw a free body diagram.

Remember that we found the value of T_{AC}. T_{AC}=33 N. Also note that to find the orange angle, we used the inverse of tan (arctan).

angle = \text{tan}^{-1}\left(\dfrac{2}{2}\right)\,=\,45^0

Let us write our equations of equilibrium for the y-axis forces.

33\text{sin}\,(45^0)\,-\,W\,=\,0

(Solve for W)

W\,=\,23.33 N

To figure out the mass, remember that W = mg.

(Where W is weight, *m* is mass, and g is the force of gravity)

m\,=\,\dfrac{23.33}{9.81}

m\,=\,2.38 kg