Solved Problem on Static Equilibrium
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For the system in equilibrium shown in the figure, determine the tension forces on the strings A
and B knowing that body C has 100 N.
Problem data:
- Weight of body C: W = 100 N.
Problem diagram:
The forces that act on the system are the gravitational force \( {\vec F}_{g} \) on block C that points down and tension forces. The cord holding the block only transmits the gravitational force of the block to the point where it is attached to the other ropes.
Cord A makes a 60° angle with the ceiling, drawing a horizontal line that passes through the point where the body is attached, we have that the tension force \( {\vec T}_{A} \) also makes a 60° angle with the horizontal, they are alternates angles.
Cord B makes a 60° angle with the vertical wall, the angle between the tension force \( {\vec T}_{B} \) and the cord that holds block C is also 60°, they are alternates angles. The angle between the horizontal line and the tension force \( {\vec T}_{B} \) is 30° with the horizontal line, they are complementary angles, adding 90°.
The forces that act on the system are the gravitational force \( {\vec F}_{g} \) on block C that points down and tension forces. The cord holding the block only transmits the gravitational force of the block to the point where it is attached to the other ropes.
Cord A makes a 60° angle with the ceiling, drawing a horizontal line that passes through the point where the body is attached, we have that the tension force \( {\vec T}_{A} \) also makes a 60° angle with the horizontal, they are alternates angles.
Cord B makes a 60° angle with the vertical wall, the angle between the tension force \( {\vec T}_{B} \) and the cord that holds block C is also 60°, they are alternates angles. The angle between the horizontal line and the tension force \( {\vec T}_{B} \) is 30° with the horizontal line, they are complementary angles, adding 90°.
Solution
First, we will decompose the forces that act in the system in coordinate system xy (Figure 2).
The gravitational force
\( {\vec F}_{g} \)
has only the component
\( {\vec F}_{gy} \)
in the negative y direction. The tension force
\( {\vec T}_{A} \)
has the component
\( {\vec T}_{Ax} \)
in the positive x direction and the component
\( {\vec T}_{Ay} \)
in the positive y direction. The tension force
\( {\vec T}_{B} \)
has the component
\( {\vec T}_{Bx} \)
in the negative x direction and the component
\( {\vec T}_{By} \)
in the negative y direction.
As the system is in equilibrium, the resultant from the forces that act on it is equal to zero.
As the system is in equilibrium, the resultant from the forces that act on it is equal to zero.
\[
\begin{gather}
\bbox[#99CCFF,10px]
{\sum \vec{F}=0}
\end{gather}
\]
Figure 2
Direction x: \( -\vec{T}_{Bx}+\vec{T}_{Ax}=0 \)
Direction y: \( -\vec{P}_{y}-\vec{T}_{By}+\vec{T}_{Ay}=0 \)
\[
\begin{gather}
-T_{B}\cos 30°+T_{A}\cos60°=0 \\[5pt]
-P-T_{B}\sin 30°+T_{A}\operatorname{sen}60°=0 \\[5pt]
\end{gather}
\]
From the Trigonometry
\[ \cos 30°=\frac{\sqrt{3}}{2} \]
\[ \cos 60°=\frac{1}{2} \]
\[ \sin 30°=\frac{1}{2} \]
\[ \sin 60°\frac{\sqrt{3}}{2} \]
These expressions can be written as a system of two equations to two unknowns, TA and TB
\[
\left\{
\begin{array}{l}
-\dfrac{\sqrt{3}}{2}T_{B}+\dfrac{1}{2}T_{A}=0 \\
-100-\dfrac{1}{2}T_{B}+\dfrac{\sqrt{3}}{2}T_{A}=0
\end{array}
\right.
\]
solving the first equation for TA
\[
\begin{gather}
\frac{1}{\cancel{2}}T_{A}=\frac{\sqrt{3}}{\cancel{2}}T_{B}\\[5pt]
T_{A}=\sqrt{3}\;T_{B} \tag{I}
\end{gather}
\]
substituting the expression (I) into the second equation of the system, we have TB
\[
\begin{gather}
-100-\frac{1}{2}T_{B}+\frac{\sqrt{3}}{2}\times \sqrt{3}\;T_{B}=0\\[5pt]
\frac{-{1}}{2}T_{B}+\frac{3}{2}T_{B}=100\\[5pt]
\frac{2}{2}T_{B}=100
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{T_{B}=100\;\text{N}}
\end{gather}
\]
substituting the value found above in the expression (I), we obtain TA
\[
\begin{gather}
T_{A}=\sqrt{3}\times 100
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{T_{A}\simeq 173\;\text{N}}
\end{gather}
\]
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .