We have a gate (3 meter long and 3 meter wide) at a 40o angle being held onto the slab by the pressure of the water with a hinge at the base. The distance z is 0.75 m. Assume that air at atmospheric pressure is above the water on the right hand side and on left hand side of the gate.
If the gate is too heavy, the force of the water would not be sufficient to keep it up. What is the limiting weight of the gate assuming that the pressure above the fluid is 1 atmosphere? (Hint: assume that the weight of the gate is exerted through its centroid.)
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How to find the net force excreted to the gate ?
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To find the limiting weight of the gate, we need to consider the equilibrium of forces acting on the gate. Since the gate is being held up by the pressure of the water, we need to calculate the maximum weight the water pressure can support.
The pressure of the water on the gate can be calculated using the concept of hydrostatic pressure. The formula for hydrostatic pressure is given by:
P = ρ * g * h
where P is the pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the depth of the fluid.
Since we are given the angle of the gate and the distance z, we can calculate the depth of the fluid using some trigonometry. Since the gate is at a 40-degree angle, the height of the fluid is given by:
h = z / sin(40)
Next, we need to calculate the pressure. The density of water is approximately 1000 kg/m^3, and the acceleration due to gravity is approximately 9.8 m/s^2. Plugging in these values, we get:
P = (1000 kg/m^3) * (9.8 m/s^2) * (z / sin(40))
Now, we need to calculate the force exerted by the water pressure on the gate. The force can be calculated using the formula:
F = P * A
where F is the force, P is the pressure, and A is the area of the gate.
The area of the gate is given by the length times the width. Plugging in the values, we get:
A = (3 m) * (3 m)
Finally, we can calculate the limiting weight of the gate by equating the force exerted by the water pressure to the weight of the gate. Assuming that the weight of the gate is exerted through its centroid, the weight can be calculated using the formula:
W = mg
where W is the weight, m is the mass, and g is the acceleration due to gravity.
Plugging in the values, we get:
W = F
W = (P * A)
W = ((1000 kg/m^3) * (9.8 m/s^2) * (z / sin(40))) * (3 m * 3 m)
Simplifying the expression, we can solve for the limiting weight of the gate.