Answer to Question #269091 in Classical Mechanics for joseph

Question

Part 1. The tide gate opens automatically when the tidewater at B subsides, allowing the marsh at A to drain. For the water level h = 4 m, determine the horizontal reaction at the smooth stop C. The gate has a width of 2 m. At what height h will the gate be on the verge of opening?

Part 2. Determine the critical height h of the water level where the concrete gravity darn starts to tip over. The density of concrete is p_c = 240 Mg/m³. Hint: Work the problem using a 1-m width of the dam.

Answer 1

Since the gate has a constant width of b = 2 m, the intensities of the distributed load on the left and right sides of the gate at C are

$(W_{C})_{L} = \rho _{w}gh_{BC} (b) = 78.48\cdot 10^{3}~N/m,$

$(W _ C ​ ) _R ​ =ρ _w ​ gh _{ AC} ​ (b) = 68.67\cdot 10^{3}~ N/m,$

the resultant triangular distributed load on the left and right sides of the gate is shown on its free-body diagram, Fig. a,

$FL = \frac{1}{2}(W_{C})_{L}L_{BC} = 156.96\cdot10^{3}~ N, \\ FR = \frac{1}{2} (W_{C})_{L}L_{AC} =120.17\cdot 10^{3}~ N,$

referring to the free-body diagram of the gate in Fig. a,

$\Sigma M_D=0,$

$F_C=25.3~kN.$