Question 6.87: The flat-bed trailer has a weight of 7000 lb and center of g...
The flat-bed trailer has a weight of 7000 lb and center of gravity at G_T . It is pin connected to the cab at D. The cab has a weight of 6000 lb and center of gravity at G_C. Determine the range of values x for the position of the 2000-lb load L so that when it is placed over the rear axle, no axle is subjected to more than 5500 lb. The load has a center of gravity at G_L.
Case 1: Assume A_y=5500 \mathrm{lb}
\hookrightarrow +\Sigma M_B=0 ; \quad-5500(13)+6000(9)+D_y(3)=0D_y=5833.33 \mathrm{lb}
+↑ \Sigma F_y=0 ; \quad B_y-6000-5833.33+5500=0
B_y=6333.33 \mathrm{lb}>5500 \mathrm{lb} (N.G!)
Case 2: Assume B_y=5500 \mathrm{lb}
\hookrightarrow +\Sigma M_A=0 ; \quad 5500(13)-6000(4)-D_y(10)=0D_y=4750 \mathrm{lb}
+↑ \Sigma F_y=0 ; \quad A_y-6000-4750+5500=0
A_y=5250 \mathrm{lb}
+↑ \Sigma F_y=0 ; \quad 4750-7000-2000+C_y=0
C_y=4250 \mathrm{lb}<5500 \mathrm{lb} (O.K!)
\hookrightarrow + \Sigma M_D=0 ; \quad-7000(13)-2000(13+12-x)+4250(25)=0x=17.4 \mathrm{ft}
Case 3: Assume C_y=5500 \mathrm{lb}
+↑ \Sigma F_y=0 ; \quad D_y-9000+5500=0
D_y=3500 \mathrm{lb}
\hookrightarrow +\Sigma M_C=0 ; \quad-3500(25)+7000(12)+2000(x)=0x=1.75 \mathrm{ft}
\hookrightarrow + \Sigma M_A=0 ; \quad-6000(4)-3500(10)+B_y(13)=0B_y=4538.46 \mathrm{lb}<5500 \mathrm{lb} (O. K!)
+↑ \Sigma F_y=0 ; \quad A_y-6000-3500+4538.46=0
A_y=4961.54 \mathrm{lb}<5500 \mathrm{lb} (O. K!)
Thus, 1.75 \mathrm{ft} \leq x \leq 17.4 \mathrm{ft}
