A gate rotates at point A and releases water when the water's depth, , exceeds a certain value. (Figure 1) The gate's width (perpendicular to the image) is = 3.00 , the water's depth is = 4.04 , and point A is located at = 2.02 . What is , the magnitude of the horizontal force exerted on the gate by the water? The water's specific weight is = 62.4 .
Your numbers require dimensions. Feet? Meters? Pounds? Slug/ft^3?
To calculate the magnitude of the horizontal force exerted on the gate by the water, we can use the following formula:
Force = Pressure * Area
First, let's calculate the pressure exerted by the water on the gate. The pressure is given by:
Pressure = ρ * g * h
where:
ρ = specific weight of water = 62.4 lb/ft³
g = acceleration due to gravity = 32.2 ft/s²
h = depth of water = 4.04 ft
Substituting the given values, we have:
Pressure = 62.4 * 32.2 * 4.04
Next, let's calculate the area of the gate. The area is given by:
Area = width * height
where:
width = 3.00 ft
height = 4.04 ft
Substituting the given values, we have:
Area = 3.00 * 4.04
Finally, we can calculate the magnitude of the horizontal force exerted on the gate by multiplying the pressure and the area:
Force = Pressure * Area
Substituting the calculated values, we have:
Force = (62.4 * 32.2 * 4.04) * (3.00 * 4.04)
Calculating this expression will give you the magnitude of the horizontal force exerted on the gate by the water.
To find the magnitude of the horizontal force exerted on the gate by the water, we can use the equation for the hydrostatic force:
F = ρghA
Where:
F is the magnitude of the force,
ρ is the density of the water,
g is the acceleration due to gravity,
h is the depth of the water,
A is the area of the gate.
First, we need to find the area of the gate. Since the width of the gate is given as 3.00 and the water's depth is given as 4.04, the area is:
A = width × depth = 3.00 × 4.04 = 12.12 square units
Next, we need to calculate the density of the water using its specific weight. The specific weight is given as 62.4. The relation between density (ρ) and specific weight (γ) is:
γ = ρg
Where γ is the specific weight and g is the acceleration due to gravity. Rearranging this equation, we can find ρ:
ρ = γ / g
Substituting γ = 62.4 and g = 9.8 (approximate value for Earth's gravity), we can calculate:
ρ = 62.4 / 9.8 = 6.3679 kg/m³ (rounded to four decimal places)
Now we have all the values needed to calculate the magnitude of the force:
F = ρghA
= 6.3679 × 9.8 × 4.04 × 12.12
≈ 3055.67 N
Therefore, the magnitude of the horizontal force exerted on the gate by the water is approximately 3055.67 N.