Consider two rectangular gates shown below; both are of the same size, but gate
ID: 1817627 • Letter: C
Question
Consider two rectangular gates shown below; both are of the same size, but gate A is held in place by a horizontal shaft through its midpoint, while gate B is held by a shaft at its top. Find the torque T required to hold the gates in place for the following scenarios: Gate A for H = 6 m and H = 8 m Gate B for H = 6 m and H = 8 m For Gate A, choose the Statement that is valid, briefly explain your answers The hydrostatic force acting on the gate increases as H increase The distance between the center of pressure and the center of gravity of the gate decreases as H increases. The torque applied to the shaft to keep the gate from turning remains constant as H increases. Assume the gate height is 4 m and width (depth in or out of the paper) is 2 m.Explanation / Answer
Lets find the force on the gate.
The pressure at a depth of (H) is P=gH
Given this the pressure at the top of the gate is: P_top = *g*(H-4)
The pressure at the bottom of the gate is: P_bottom = *g*H
What this gives you is a pressure gradient that is shaped like:
/---------|
/ |
/ |
/ |
/____________|
with a base of *g*H a top of *g*(H-4).
The shape can be thought of a triangle stacked ontop of a rectangle. We know that the force of a uniform distribution (rectange) acts at 1/2 its height. The force on a linear varring distribution (triangle) is located at 1/3 its height.
The dimensions of the rectange are 4m x *g*(H-4).
The dimensions of the triangle are 4m x *g*4.
Force from rectange = 4*2**g*(H-4) = 8**g*(H-4)
Force from Triangle = (1/2)*4**g*H*2*4 = 16**g*H
Gate B: T = [8*p*g*(H-4) * 2] + [16*p*g*H *(8/3)]
Gate A: T = 16*p*g*H*(2-(4/3))
a.) H = 6m: T = 16*1*9.8*6*(2/3) = 627.2 Nm
H = 8m: T = 16*1*9.8*8*(2/3) = 836.27 Nm
b.) H = 6m: T = [8*1*9.8*2*2]+[16*1*9.8*6*(8/3)] = 2822.4 Nm
H = 8m: T = [8*1*9.8*4*2]+[16*1*9.8*8*(8/3)] = 3972.27 Nm
c.) The first statement is true, as the equation above shows the torque is proportional to the pressure which is proportional to the height of the water column.