To solve these parts, let's break down the question into the components provided:

Part (a)

We need to find the hydrostatic force $f(y)$ exerted by the water on the rectangular door between the height $y$ and $y + \Delta y$.

Hydrostatic Force Formula

The hydrostatic force on a small area at a depth is given by the formula: $f(y) = P \cdot A$ where:

• $P$ is the pressure at a depth $y$,
• $A$ is the area of the door segment at depth $y$.

Pressure at Depth $y$

The pressure at depth $y$ below the surface of the water is given by the hydrostatic pressure formula: $P = \rho g y$ where:

• $\rho = 1000 \, \text{kg/m}^3$ is the density of water,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $y$ is the depth from the surface of the water.

Area of the Door Segment

The area $A$ of the door segment between $y$ and $y + \Delta y$ is the width of the door (which is 4 meters) multiplied by the small height $\Delta y$: $A = 4 \, \text{m} \times \Delta y$

Force $f(y)$

Now, the force $f(y)$ exerted on the small segment is: $f(y) = P \cdot A = (\rho g y) \cdot (4 \Delta y)$ Substitute the known values for $\rho$ and $g$: $f(y) = (1000 \times 9.8 \times y) \cdot (4 \Delta y) = 39200 y \Delta y$ Thus, the expression for the hydrostatic force is: $f(y) = 39200 y \Delta y \, \text{N}$

Part (b)

Now, we need to calculate the total hydrostatic force exerted on the entire door. This is obtained by integrating the force over the height of the door, from $y = 0$ (at the base) to $y = 3$ meters (the top of the door).

Total Force

The total hydrostatic force $F$ is the integral of $f(y)$ from $y = 0$ to $y = 3$: $F = \int_0^3 39200 y \, dy$ To calculate this: $F = 39200 \int_0^3 y \, dy = 39200 \left[ \frac{y^2}{2} \right]_0^3$ $F = 39200 \times \frac{9}{2} = 39200 \times 4.5 = 176400 \, \text{N}$

Final Answers:

• (a) The hydrostatic force exerted on the door between $y$ and $y + \Delta y$ is: $f(y) = 39200 y \Delta y \, \text{N}$
• (b) The total hydrostatic force exerted on the door is: $F = 176400 \, \text{N}$

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Would you like more details on the steps or have any questions about the calculations? Here are some related questions:

1. What is the effect of increasing the width of the door on the total hydrostatic force?
2. How would the force change if the water was deeper, say 20 m instead of 10 m?
3. What happens to the force if the door were positioned at different heights from the base of the dam?
4. How would the hydrostatic force be affected if the density of water were to change (e.g., salty water)?
5. How is the hydrostatic pressure formula derived in terms of fluid statics?

Tip: Remember that the force increases linearly with depth, so the bottom of the door experiences the highest pressure!