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.

Determine the mass of each of the two cylinders

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.

Determine the mass of each of the two cylinders

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

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-17.

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