Problem $5$: In class, it was shown that a linearly distributed loading can be simplified to an equivalent

system that consists of an equivalent resultant force $P_R$ acting at the centroid $C$ of the triangular loading

distribution. The location of the centroid was specified as $\frac{2}{3}L$ measured from the lower end of the

triangular loading, where $L$ is the length over which the linearly distributed loading acts. Use the general

equations for the magnitude and location of the resultant force of a distributed load $($integral equations$)$ to

prove that the location of the centroid for a linearly distributed loading is at a distance $\frac{2}{3}L.$ In the diagram

below, $w$ is the value of the triangular load at its highest point and has units of force$/$length$.$

$5\mathrm{.}1)$ Determine the value of the distributed load as a function of $x,$ measured from the lower end of the

triangular loading $($i$.$e$.,w(x)).$

$5\mathrm{.}2)$ Use the general equation for the magnitude of a distributed loading to determine the value of the

resultant force $P_R.$

$5\mathrm{.}3)$ Use the general equation for the location of the resultant force of a distributed loading to determine

the distance to the centroid $\stackrel{}{x}.$