Question

The three ropes in FIGURE are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. What is the magnitude of the tension in the third rope? Use a=0.63 m, b=0.57 m, T1=39.7 N, T2=91.1 N.
What is the direction of the tension in the third rope? (relative to horizontal)

Answer 1

in this problem due to the very light of the ring we will ignore weight of the ring because teh ring is in equlibrium it must obey \(\mathrm{F}_{n d}=0\)

so it has both \(\mathrm{x}\) - components and \(y\) - components \(\left(F_{\text {st }}\right)_{x}=T_{3} \cos \theta-T_{1}=0 N\)

\(T_{3} \cos \theta=T_{3}\)

\(\left(F_{\mathrm{w}_{7}}\right)_{y}=T_{1}-T_{3} \sin \theta=0 \mathrm{~N}\)

\(T_{3} \sin \theta=T_{2}\)

\(\frac{T_{3} \sin \theta}{I_{3} \cos \theta}=\tan \theta\)

\(\frac{T_{2}}{T_{1}}=\frac{91.1}{39.7}=2.29\)

\(\theta=\tan ^{-1}(2.29)=66.45^{\circ}\)

the tension in the rope

\(\mathrm{T}_{3}=\frac{T_{1}}{\cos 66.45^{\circ}}\)

\(\frac{39.47}{\cos 66.45^{\circ}}=98.92 \mathrm{~N}\)

the tension in the third rope is \(98.92 \mathrm{~N}\) directed \(66.45^{\circ}\) below the horizontal.