7-7. Determine the internal normal force, shear force, and moment at point \( C \) in the cantilever beam.
The intensity of the triangular distributed loading at \( C \) can be computed using the similar triangles shown in Fig. \( a \),
\[
\frac{w_{C}}{L / 2}=\frac{w_{0}}{L} \text { or } w_{C}=w_{0} / 2
\]
With reference to Fig. \( b \),
\[
\begin{array}{lll}
\stackrel{+}{\rightarrow} \Sigma F_{x}=0 & N_{C}=0 \\
+\uparrow \Sigma F_{y}=0 ; & V_{C}-\left(\frac{w_{0}}{2}\right)\left(\frac{L}{2}\right)-\frac{1}{2}\left(\frac{w_{0}}{2}\right)\left(\frac{L}{2}\right)=0 & V_{C}=\frac{3 w_{0} L}{8} \\
+\Sigma M_{C}=0 ; & -M_{C}-\left(\frac{w_{0}}{2}\right)\left(\frac{L}{2}\right)\left(\frac{L}{4}\right)-\frac{1}{2}\left(\frac{w_{0}}{2}\right)\left(\frac{L}{2}\right)\left(\frac{L}{3}\right)=0 & M_{C}=-\frac{5}{48} w_{0} L^{2}
\end{array}
\]
The negative sign indicates that \( \mathbf{M}_{C} \) acts in the opposite sense to that shown on the free-body diagram.
(a)
(b)