Question

A gate is held closed by a horizontal force, F, located at the center of the gate. The maximum value for F is 4500 kN. Determine the maximum water depth h above the center of the gate that can exist without the gate opening?

Answer 1

To find the maximum water depth that can exist without opening the gate, use the formula
`F = pgh²L/2`, rearrange it to solve for h, and input the known values including the maximum force F, density of water p, gravity g, and the dam length L.

Step-by-step explanation:

To determine the maximum water depth h above the center of the gate that can exist without the gate opening, we must use the equation that describes the force exerted on a dam by the water behind it. According to physics, this force can be calculated using the formula
`F = pgh²L/2`, where p is the density of water, g is the acceleration due to gravity, h is the depth of the water at the dam, and L is the length of the dam. Given that the maximum force F is 4500 kN, we can rearrange the formula to solve for h:

`h = \sqrt(2F)/(pgL))`

To find the value of h, substitute F = 4500 kN (which is 4500 x 10³ N), the density of water p = 1000 kg/m³ (assuming room temperature and fresh water), the acceleration due to gravity g = 9.81 m/s², and the length of the dam L. The square root of the resulting calculation gives the maximum depth of water the gate can hold without opening. Note that the length of the dam, L, must be known to complete this calculation.