Related questions
Question
thumb_up100%
![**Hydrostatic Force on a Submerged Circular Plate**
A circular plate with a radius of 9 meters is submerged vertically in water as illustrated in the diagram below. We are required to express the hydrostatic force against one side of the plate as an integral and evaluate it. Your answer should be rounded to the nearest whole number. Use 9.8 m/s² for the acceleration due to gravity. Recall that the mass density of water is 1000 kg/m³.
### Integral Expression
The hydrostatic force \( F \) can be expressed as:
\[
F = \rho g \int_{-9}^{0} (12 - y) \sqrt{81 - y^2} \, dy
\]
Where:
- \(\rho\) is the mass density of water, \(1000 \, \text{kg/m}^3\).
- \(g\) is the acceleration due to gravity, \(9.8 \, \text{m/s}^2\).
- \( (12 - y) \) represents the depth of the submerged section.
- \( \sqrt{81 - y^2} \) is derived from the circle's equation for the half-chord length at depth \( y \).
### Calculation
Evaluating the integral gives the hydrostatic force:
\[
F \approx 14,962,777 \, \text{N}
\]
### Diagram Description
The diagram shows a circle submerged in water. The radius of the circle is 9 meters, and it is vertically positioned with a center depth of 12 meters from the surface. A vertical line through the center is annotated, extending 3 meters above the top of the circle to indicate the depth from the water's surface.
This setup helps visualize the distribution of pressure across the submerged plate, informing calculations for hydrostatic force.](https://content.bartleby.com/qna-images/question/e7d29c3c-46d0-4481-8869-c2379eb7373c/44f63084-c9ee-4781-acdc-84f8e921f389/5d2pe1e_processed.jpeg)
Transcribed Image Text:**Hydrostatic Force on a Submerged Circular Plate**
A circular plate with a radius of 9 meters is submerged vertically in water as illustrated in the diagram below. We are required to express the hydrostatic force against one side of the plate as an integral and evaluate it. Your answer should be rounded to the nearest whole number. Use 9.8 m/s² for the acceleration due to gravity. Recall that the mass density of water is 1000 kg/m³.
### Integral Expression
The hydrostatic force \( F \) can be expressed as:
\[
F = \rho g \int_{-9}^{0} (12 - y) \sqrt{81 - y^2} \, dy
\]
Where:
- \(\rho\) is the mass density of water, \(1000 \, \text{kg/m}^3\).
- \(g\) is the acceleration due to gravity, \(9.8 \, \text{m/s}^2\).
- \( (12 - y) \) represents the depth of the submerged section.
- \( \sqrt{81 - y^2} \) is derived from the circle's equation for the half-chord length at depth \( y \).
### Calculation
Evaluating the integral gives the hydrostatic force:
\[
F \approx 14,962,777 \, \text{N}
\]
### Diagram Description
The diagram shows a circle submerged in water. The radius of the circle is 9 meters, and it is vertically positioned with a center depth of 12 meters from the surface. A vertical line through the center is annotated, extending 3 meters above the top of the circle to indicate the depth from the water's surface.
This setup helps visualize the distribution of pressure across the submerged plate, informing calculations for hydrostatic force.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps with 1 images
Knowledge Booster
Similar questions
- A vertical plate is partially submerged in water and has the indicated shape. 12 m pg 4 m Express the hydrostatic force (in N) against one side of the plate as an integral (let the positive direction be upwards) and evaluate it. (Round your answer to the nearest whole number. Use 9.8 m/s2 for the acceleration due to gravity. Recall that the weight density of water is 1,000 kg/m³.) x ) dy = [ D Narrow_forwardA trough is filled with milk with a liquid of density of 64.6 pounds per cubic foot. The ends of the trough are equilateral triangles with sides 8 Feet long and vertex at the bottom. Find the hydrostatic force on one end of the trough.arrow_forward:A vertical plate is submerged in water and has the indicated shape. 7 ft 6 ft 5 ft 5 ft Express the hydrostatic force (in ft) against one side of the plate as an integral (let the positive direction be downwards) and evaluate it. (Recall that the weight density of water is 62.5 Ib/ft°.) dx = ftarrow_forward
- garbage worker tears the arm off a droid. he throws it down a garbage chute and it lands in trash filled with nasty smelly liquids. the overall density is 840kg/m^3. if the arm has a desnity of 5460kg/m^3 and is a cylinder with the volume of 0.0330m^3 what is the buyoncy force in newtons acting on the arm?arrow_forwardIf this diver has a mass of 150 kg ( including his diver gear), and a total volume of 0.14 m3 , How much force is provided by the bottom of the swimming pool on which he is sitting?arrow_forwardA geologist finds that a moon rock whose mass is 8.25 kg has an apparent mass of 6.15 kg when submerged in water. What is the density of the moon rock? Given that the density of water is 1,000 kg/m³. Include a diagram of the situation.arrow_forward
arrow_back_ios
arrow_forward_ios