We need to find the angle that force T makes with the positive x-axis first. We call this the angle beta, β. This is depicted in the free-body diagram.
Free-body diagram:
Solving for the values of angles α and β.
\begin{aligned}
\tan \alpha & = \dfrac{3}{4} \\
\alpha & = \tan ^{-1} \frac{3}{4} \\
\alpha & = 36.8699 \degree \\
\end{aligned}Knowing that the sum of angles α and β is 90°, we can solve for the β.
\begin{aligned}
\alpha + \beta & = 90\degree \\
\beta & = 90 \degree - \alpha \\
\beta & = 90 \degree - 36.8699 \degree \\
\beta & = 53.1301 \degree
\end{aligned}Equations of Equilibrium:
Summation of forces in the x-direction:
\begin{aligned}
\xrightarrow{+} \sum F_x & = 0 \\
T \cos \beta - \frac{4}{5} F & = 0 \\
T \cos 53.1301 \degree - \frac{4}{5} F & = 0 & & \qquad \qquad (1)\\
\end{aligned}Summation of forces in the y-direction:
\begin{aligned}
+\uparrow \sum F_y & =0 \\
9 - \frac{3}{5} F- T \sin \beta & = 0 \\
T \sin 53.1301 \degree + \frac{3}{5}F & = 9 & & \qquad \qquad(2)\\
\end{aligned}Now, we have two equations with two unknowns. We shall solve the unknowns by solving these equations simultaneously. We can use our calculator, or we can solve this manually using the method of substitution.
Using equation (1), solve for T in terms of F.
\begin{aligned}
T \cos 53.1301\degree-\frac{4}{5} F & = 0 \\
T \cos 53.1301\degree & = \frac{4}{5} F \\
T & = \dfrac{\frac{4}{5}F}{\cos 53.1301\degree} \qquad \qquad (3)\\
\end{aligned}Now, substitute this equation (3) to equation (2) to solve for F:
\begin{aligned}
T \sin 53.1301 \degree + \frac{3}{5}F & = 9 \\
\left(\dfrac{\frac{4}{5}F}{\cos 53.1301\degree} \right) \sin 53.1301 \degree + \frac{3}{5}F & = 9 \\
\frac{4}{5}F \left( \dfrac{\sin 53.1301\degree}{\cos 53.1301\degree}\right)+ \frac{3}{5}F & = 9 \\
\frac{4}{5}F \tan 53.1301\degree+\frac{3}{5}F &=9 \\
F\left( \frac{4}{5} \cdot \tan 53.1301\degree +\frac{3}{5}\right) & = 9\\
F & = \dfrac{9}{\frac{4}{5} \cdot \tan 53.1301\degree +\frac{3}{5}} \\
F & = 5.4 \ \text{kN} \\
\end{aligned}Substitute the value of F to equation (3) to solve for T:
\begin{aligned}
T & = \dfrac{\frac{4}{5}F}{\cos 53.1301\degree} \\
T & = \dfrac{\frac{4}{5} \cdot \left( 5.4 \ \text{kN}\right)}{\cos 53.1301\degree} \\
T & = 7.2 \ \text{kN}
\end{aligned}Therefore, F = 5.4 \ \text{kN} and T= 7.2 \ \text{kN} .
Free-body diagram of the roller:
Equations of Equilibrium:
Note that if we take the sum of forces in the x-direction, there are two unknown forces involve, but if we take the sum of forces in the y-direction, there is only one unknown force involve.
Summation of forces in the y-direction:
\begin{aligned}
+\uparrow \sum F_y & =0& & & & & \\
125- N_C \cos 40 \degree &=0 & & & & &\\
N_C &=\dfrac{125}{\cos 40 \degree} & & & & & \\
N_C & =163.1759 \ \text{N} \\
\end{aligned}Summation of forces in the x-direction:
\begin{aligned}
\xrightarrow{+} \sum F_x & =0 \\
N_B - 163.1759\ \sin 40 \degree &=0 \\
N_B &=163.1759 \sin 40\degree \\
N_B & = 104.8874 \ \text{N}
\end{aligned}Therefore, the normal reactions NB and NC on the bearing at its contact points B and C for equilibrium are 163.1759 N and 104.8874 N, respectively.
Free-body Diagram:
Equilibrium Equation:
Summation of forces in the x-direction:
\begin{aligned}
\xrightarrow{+} \: \sum F_x & = 0 & \\
5 \: \text{kN}+F \sin \theta - 8 \: \text {kN} \cos 30 \degree - 4\: \text{kN} \cos 60 \degree & = 0 & \\
F \sin \theta &= 3.9282 & (1)
\end{aligned}Summation of forces in the y-direction:
\begin{aligned}
+\uparrow \sum F_y & = 0 &\\
8 \sin 30 \degree - 4 \sin 60 \degree - F \cos \theta & =0 &\\
F \cos \theta & = 0.5359 & (2)\\
\end{aligned}We now have two equations. Divide Eq (1) by (2)
\begin{aligned}
\dfrac{F \sin \theta}{F \cos \theta} &= \dfrac{3.9282}{0.5359} \\
\dfrac{ \sin \theta}{ \cos \theta} & = 7.3301 \\
\end{aligned}We know that \tan \theta = \dfrac{\sin \theta}{\cos \theta} :
\begin{aligned}
\tan \theta &=7.3301 \\
\theta & = \tan^{-1}7.3301\\
\textcolor{blue}\theta & \textcolor{blue}{=82.2\degree}\\
\end{aligned}Substituting this result to equation (1), we have
\begin{aligned}
F\sin 82.2 \degree & = 3.9282 \\
\textcolor{blue}F & \textcolor{blue}{=3.96 \ \text{kN}}
\end{aligned}
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