A sluice gate consists of a quadrant of a circle of radius 1.5m pivoted at its centre O. Its centre of gravity is at G, as Shown in fig 1. When the water is level with the pivot O, calculate the magnitude and direction of the resultant force on the gate due to the water and the turning moment required to open the gate. The width of the gate is 3m and it has a mass of 6000kg.
We can begin by calculating the force due to the water pressure acting on the gate. The pressure at a depth h under the surface of the water is given by P = ρgh, where ρ is the density of water (1000 kg/m³), g is the acceleration due to gravity (9.81 m/s²), and h is the depth of the centroid of the gate.
The centroid of the quadrant of the circle lies at a radius of 4/3*pi from the center (you can calculate this using the formula for the centroid of a curve: (1/(pi R))*(integral from 0 to R of (R^2 - x^2)dx)). Therefore, the depth of the centroid is h = 1.5m - (4/3*pi).
Now we can calculate the pressure acting on the gate at the centroid:
P = ρgh = (1000 kg/m³)(9.81 m/s²)(1.5m - (4/3*pi))
P ≈ 6767.77 N/m²
The total force due to the water pressure is given by F = PA, where A is the area of the gate. Since the gate is a quadrant of a circle with a radius of 1.5m, its area is given by A = 1/4 * pi * (1.5m)² = 1.7671 * pi m². The width of the gate is 3m, so the total volume of the gate is 1.7671 * pi * 3 = 16.621 m². The total force due to the water pressure is then:
F = PA = (6767.77 N/m²)(16.621 m²) = 112540.23 N
The turning moment required to open the gate depends on the weight of the gate itself. Since the gate has a mass of 6000 kg, its weight is given by W = mg = (6000 kg)(9.81 m/s²) = 58860 N.
The weight of the gate acts downward at the centroid G, creating a clockwise turning moment about the pivot O. The magnitude of this turning moment is given by M = rW, where r is the perpendicular distance from the line of action of the force to the pivot. Since the gate is a quadrant of a circle, the distance between O and G is (4/3/pi)(1.5m) ≈ 0.6366 m.
Thus, the magnitude of the turning moment due to the weight of the gate is M = rW = (0.6366 m)(58860 N) = 37432.996 Nm.
The force due to the water pressure on the gate acts at a perpendicular distance of 2r/3 from the pivot O. Therefore, the magnitude of the turning moment due to the water pressure is M' = r'F = (2/3*r)(112540.23 N) = (2/3 * 1.5m)(112540.23 N) = 112540.23 Nm.
The net turning moment required to open the gate is the difference between the two moments:
Net turning moment = M' - M = (112540.23 Nm) - (37432.996 Nm) = 75107.234 Nm
The force exerted at the centroid is parallelogram, comprising the hydrostatic force and the vertical force due to the weight of gate.
Magnitude of the Resultant Force = sqrt(F^2 + W^2)
Magnitude of the Resultant Force = sqrt((112540.23 N)^2 + (58860 N)^2) = 125000 N (approximately)
The direction of the Resultant Force can be found using the tangent of the angle between the horizontal force and the Resultant Force:
tan(angle) = W / F
angle = arctan(W / F)
angle = arctan(58860 N / 112540.23 N) = 27.5 degrees (approximately)
So, the magnitude of the resultant force on the gate due to the water is approximately 125000 N and the direction of the force is 27.5 degrees from the horizontal. The turning moment required to open the gate is approximately 75107.234 Nm.
To calculate the magnitude and direction of the resultant force on the gate due to the water, we need to analyze the forces acting on the gate.
1. Weight of the gate:
The weight of the gate can be calculated using the formula:
Weight = Mass x gravitational acceleration
Given that the mass of the gate is 6000 kg and the gravitational acceleration is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 6000 kg x 9.8 m/s^2 = 58800 N
2. Buoyancy force:
The buoyancy force acts vertically upwards on the gate and is equal to the weight of the water displaced by the gate when it is submerged. Since the water is level with the pivot O, the entire quadrant of the circle is submerged, creating a semicircular shape. The volume of a semicircle can be calculated using the formula:
Volume = (1/2) x π x r^2
In this case, the radius is 1.5 m, so the volume is:
Volume = (1/2) x π x (1.5 m)^2 = 3.53 m^3
The density of water is approximately 1000 kg/m^3, so the mass of the displaced water is:
Mass of displaced water = Volume x Density = 3.53 m^3 x 1000 kg/m^3 = 3530 kg
The buoyancy force can be calculated using the formula:
Buoyancy force = Mass of displaced water x gravitational acceleration
Buoyancy force = 3530 kg x 9.8 m/s^2 = 34574 N
3. Resultant force:
The resultant force is the vector sum of the weight and buoyancy force. Since the weight acts vertically downwards and the buoyancy force acts vertically upwards, the resultant force will have a vertical component equal to the difference between the two forces. The magnitude of the resultant force can be calculated as follows:
Resultant force = |Weight| - |Buoyancy force| = 58800 N - 34574 N = 24226 N
The direction of the resultant force will be upward.
To calculate the turning moment required to open the gate, we need to consider the lever arm and the force acting on it.
4. Lever arm:
The lever arm is the perpendicular distance between the pivot (O) and the line of action of the resultant force.
In this case, the lever arm can be found by taking the moment about the pivot O. Since the gate is symmetrical and the weight and buoyancy forces act along lines of symmetry, the lever arm will be the radius of the quadrant (1.5 m).
5. Turning moment:
The turning moment can be calculated using the formula:
Turning moment = Resultant force x Lever arm
Turning moment = 24226 N x 1.5 m = 36339 Nm
Therefore, the magnitude of the resultant force on the gate due to the water is 24226 N, acting upward. The turning moment required to open the gate is 36339 Nm.
To calculate the magnitude and direction of the resultant force on the gate due to the water, we need to consider the pressure of the water acting on the gate.
1. Find the area of the gate:
Since the sluice gate consists of a quadrant of a circle, the area can be calculated as a fraction of the area of a full circle with the same radius.
Area of the full circle = π * (1.5m)^2
Area of the gate = (1/4) * Area of the full circle
2. Calculate the pressure of the water on the gate:
Pressure = Force / Area
Since the water is level with the pivot, the pressure at the pivot is equal to atmospheric pressure (assumed to be 1 atm or 101325 Pa). Therefore, the pressure at any point on the gate can be assumed to be the same as the atmospheric pressure.
3. Calculate the force exerted by the water on the gate:
Force = Pressure * Area
4. Determine the direction of the force:
The force exerted by the water on the gate will act vertically upwards from the bottom of the gate.
5. Calculate the magnitude of the force:
Magnitude of force = Force
Now, let's calculate the values:
1. Area of the gate:
Area of the full circle = π * (1.5m)^2
Area of the gate = (1/4) * Area of the full circle
2. Pressure of the water on the gate:
Pressure = 1 atm = 101325 Pa
3. Force exerted by the water on the gate:
Force = Pressure * Area
4. Direction of the force:
Upwards, from the bottom of the gate.
5. Magnitude of the force:
Magnitude of force = Force
To calculate the turning moment required to open the gate, we need to consider the gravitational force acting on the gate.
6. Calculate the weight of the gate:
Weight = mass * acceleration due to gravity
7. Determine the distance between the pivot and the center of gravity of the gate (OG).
8. Calculate the turning moment required:
Turning moment = Weight * OG
Now, let's calculate the values:
6. Weight of the gate:
Weight = mass * acceleration due to gravity
7. Distance between the pivot and the center of gravity (OG):
Since the gate is a quadrant of a circle, the center of gravity will be at 2/3 of the radius from the pivot.
OG = (2/3) * radius
8. Turning moment required:
Turning moment = Weight * OG
Once you have calculated the magnitude and direction of the resultant force on the gate due to the water, as well as the turning moment required to open the gate, you will have the answers to the given problem.