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 = 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
I have a question here why we dont calculate the both spring for calculate the mass of a cyclinder ?
There isn’t a need to since it’s symmetrical.