The device shown is used to straighten the frames of wrecked autos. Determine the tension of each segment of the chain, i.e., AB and BC, if the force which the hydraulic cylinder DB exerts on point B is 3.50 kN, as shown.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
Let us first draw our free body diagram. The angles have been calculated in the diagram, but steps to calculating those angles will be shown below.
To figure out the angles, we will use simple trigonometry.
\text{tan}^{-1}(\frac{400}{450})=41.6^0
The brown angle is found by:
\text{tan}^{-1}(\frac{250}{450})=29^0
We will assume forces going \rightarrow^+ to be positive and \uparrow+ to be positive.
First, we will write equations of equilibrium for the y-axis forces as this will give us a solution to F_{BC}.
3.5\text{cos}\,(41.6^0)-F_{BC}\text{cos}\,(29^0)=0
(Solve for F_{BC})
F_{BC}=2.99 kN
Now, we will write equations of equilibrium for the x-axis forces.
\sum \text{F}_\text{x}=0
3.5\text{sin}\,(41.6^0)+2.99\text{sin}\,(29^0)-F_{AB}=0
(Substituted the value of F_{BC} we found earlier. Solve for F_{AB}.)
F_{AB}=3.77 kN
This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-7.
perfect