# Determine the mass of each of the two cylinders

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.

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

spring length = $\sqrt{2^2+1.5^2}$

spring length = $2.5$ m

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

spring length = $\sqrt{2^2+2^2}$

(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:

$F=ks$

(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.

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

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

(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)

$m\,=\,\dfrac{W}{g}$

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

$m\,=\,2.38$ kg