Question

Determine the magnitude of the resultant hydrostatic force acting on the dam and its location, measured from the top surface of the water. The width of the dam is 8 m; rhow = 1.0 Mg/m3.

Answer 1

The magnitude of the resultant hydrostatic force on the dam is approximately 470,400 N, and its location is around 12 m from the top surface of the water, indicating the center of pressure.

To determine the magnitude of the resultant hydrostatic force acting on the dam and its location, we can use the following formulas:

Magnitude of the hydrostatic force (F):

`\[ F = \rho \cdot g \cdot A \]`

where:

-
`\( \rho \)` is the density of water (given as
`\(1.0 \, \text{Mg/m}^3\)`),

- g is the acceleration due to gravity (
`\(9.8 \, \text{m/s}^2\)`),

- A is the area of the submerged surface.

Location of the hydrostatic force x:

`\[ x = (I_G)/(A) \]`

where:

`I_G` is the moment of inertia of the submerged surface about the water surface.

For a rectangular dam, the area A is given by:

`\[ A = \text{width} * \text{height} \]`

and the moment of inertia I_G about the base is given by:

`\[ I_G = (1)/(3) * \text{width} * \text{height}^3 \]`

Let's calculate these values:

Calculate the area A:

`\[ A = \text{width} * \text{height} \] \[ A = 8 \, \text{m} * 6 \, \text{m} \] \[ A = 48 \, \text{m}^2 \]`

Calculate the moment of inertia

`\[ I_G = (1)/(3) * \text{width} * \text{height}^3 \] \[ I_G = (1)/(3) * 8 \, \text{m} * (6 \, \text{m})^3 \] \[ I_G = (1)/(3) * 8 \, \text{m} * 216 \, \text{m}^3 \] \[ I_G = 576 \, \text{m}^4 \]`

Calculate the magnitude of the hydrostatic force

`\[ F = \rho \cdot g \cdot A \] \[ F = (1.0 \, \text{Mg/m}^3) * (9.8 \, \text{m/s}^2) * 48 \, \text{m}^2 \]`

`\(1 \, \text{Mg} = 1000 \, \text{kg}\)`

`\[ F = 1.0 * 1000 * 9.8 * 48 \]`

Now, you can calculate the numerical value for F and then determine the location x using the formula
`\( x = (I_G)/(A) \)`.

`\[ F = 1.0 * 1000 * 9.8 * 48 \]\[ F \approx 470,400 \, \text{N} \]`

Now, let's calculate the location x of the hydrostatic force using the formula:

`\[ x = (I_G)/(A) \]\[ x = \frac{576 \, \text{m}^4}{48 \, \text{m}^2} \]\[ x \approx 12 \, \text{m} \]`

So, the magnitude of the resultant hydrostatic force acting on the dam is approximately 470,400 N, and its location is about 12 m measured from the top surface of the water. This means the center of pressure is 12 m below the water surface.

The complete question is:

Determine the magnitude of the resultant hydrostatic force acting on the dam and its location, measured from the top surface of the water. The width of the dam is 8 m; rhow = 1.0 Mg/m3.

Answer 2

In your question a figure of dam is missing.where height is given which 6 m

Answer:

Force=1.41×10⁶ Newton

Location from surface of water= 4 m

Step-by-step explanation:

Given Data

w (width)=8 m

p=1.0 Mg/m³

Force=?

Location=?

Solution

`p=6(1)(10^(3) )(9.8)\\p=58860 (N)/(m^(2) )\\ Force=(1)/(2)(6)(8)(58,860)\\ F=1.41*10^(6)N\\ F=1.41MN\\h(location)=2/3(6)\\h=4m`