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A heavy hollow cone of radius R and height h is placed on a horizontal table surface, with its flat base on the table. The whole volume inside the cone is filled with water of density r. The circular rim of the cone's base has a watertight seal with the table's surface and the top apex of the cone has a small hole. Neglecting atmospheric pressure find the total upward force exerted by water on the cone is
- a)(2/3)pR2hrg
- b)(1/3)pR2hrg
- c)pR2hrg
- d)None
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A heavy hollow cone of radius R and height h is placed on a horizontal...
Total force exerted on the base by water and cone’s slant surface = mg = 1/3 πR2 Hρg downwards
Force exerted by the water = (ρgH) (πR2) (downwards)
So force exerted by the slant surface = 2/3 ρgH πR2 (upwards)
So force exerted by water on slant surface = 2/3 ρgHπR2
Force exerted by the water = (ρgH) (πR2) (downwards)
So force exerted by the slant surface = 2/3 ρgH πR2 (upwards)
So force exerted by water on slant surface = 2/3 ρgHπR2
Most Upvoted Answer
A heavy hollow cone of radius R and height h is placed on a horizontal...
The total upward force exerted by the water on the cone can be found by considering the pressure exerted by the water at different points on the cone's surface.
1. Pressure at the base of the cone:
- The water at the base of the cone is at the same level as the table surface, so the pressure at the base is equal to the atmospheric pressure.
- Since the circular rim of the cone's base has a watertight seal with the table surface, the water does not exert any additional upward force on the cone at the base. Therefore, the pressure at the base does not contribute to the total upward force.
2. Pressure at the top of the cone:
- The water at the top of the cone is exposed to atmospheric pressure through the small hole at the apex.
- The pressure at the top is equal to the atmospheric pressure.
3. Pressure at different points on the curved surface of the cone:
- The pressure at any point on the curved surface of the cone can be calculated using the hydrostatic pressure formula: Pressure = density × gravity × height.
- The height at any point on the curved surface can be expressed in terms of the radius and height of the cone using similar triangles: h' = (r/R) × h, where h' is the height of the water column at that point.
- The pressure at any point on the curved surface can be calculated as: Pressure = density × gravity × h'.
- The upward force exerted by the water at any point on the curved surface can be calculated as: Force = Pressure × area = density × gravity × h' × πr^2.
- To find the total upward force exerted by the water on the cone, we need to integrate the force over the entire curved surface.
- Integrating the force over the curved surface, we get: Total force = ∫(density × gravity × h' × πr^2) dA = density × gravity × πR^2 ∫(h'/R^2) dA.
- Since the volume inside the cone is completely filled with water, the integral of (h'/R^2) over the curved surface is equal to the volume of the cone divided by the area of the base of the cone.
- The volume of the cone can be calculated as: Volume = (1/3)πR^2h.
- Therefore, the total upward force exerted by the water on the cone is: Total force = density × gravity × πR^2 × (1/3)πR^2h / πR^2 = (1/3)density × gravity × R^2h.
Hence, the correct answer is option (a) (2/3)pR^2hrg.
1. Pressure at the base of the cone:
- The water at the base of the cone is at the same level as the table surface, so the pressure at the base is equal to the atmospheric pressure.
- Since the circular rim of the cone's base has a watertight seal with the table surface, the water does not exert any additional upward force on the cone at the base. Therefore, the pressure at the base does not contribute to the total upward force.
2. Pressure at the top of the cone:
- The water at the top of the cone is exposed to atmospheric pressure through the small hole at the apex.
- The pressure at the top is equal to the atmospheric pressure.
3. Pressure at different points on the curved surface of the cone:
- The pressure at any point on the curved surface of the cone can be calculated using the hydrostatic pressure formula: Pressure = density × gravity × height.
- The height at any point on the curved surface can be expressed in terms of the radius and height of the cone using similar triangles: h' = (r/R) × h, where h' is the height of the water column at that point.
- The pressure at any point on the curved surface can be calculated as: Pressure = density × gravity × h'.
- The upward force exerted by the water at any point on the curved surface can be calculated as: Force = Pressure × area = density × gravity × h' × πr^2.
- To find the total upward force exerted by the water on the cone, we need to integrate the force over the entire curved surface.
- Integrating the force over the curved surface, we get: Total force = ∫(density × gravity × h' × πr^2) dA = density × gravity × πR^2 ∫(h'/R^2) dA.
- Since the volume inside the cone is completely filled with water, the integral of (h'/R^2) over the curved surface is equal to the volume of the cone divided by the area of the base of the cone.
- The volume of the cone can be calculated as: Volume = (1/3)πR^2h.
- Therefore, the total upward force exerted by the water on the cone is: Total force = density × gravity × πR^2 × (1/3)πR^2h / πR^2 = (1/3)density × gravity × R^2h.
Hence, the correct answer is option (a) (2/3)pR^2hrg.
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A heavy hollow cone of radius R and height h is placed on a horizontal table surface, with its flat base on the table. The whole volume inside the cone is filled with water of density r. The circular rim of the cone's base has a watertight seal with the table's surface and the top apex of the cone has a small hole. Neglecting atmospheric pressure find the total upward force exerted by water on the cone isa)(2/3)pR2hrgb)(1/3)pR2hrgc)pR2hrgd)NoneCorrect answer is option 'A'. Can you explain this answer?
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A heavy hollow cone of radius R and height h is placed on a horizontal table surface, with its flat base on the table. The whole volume inside the cone is filled with water of density r. The circular rim of the cone's base has a watertight seal with the table's surface and the top apex of the cone has a small hole. Neglecting atmospheric pressure find the total upward force exerted by water on the cone isa)(2/3)pR2hrgb)(1/3)pR2hrgc)pR2hrgd)NoneCorrect answer is option 'A'. Can you explain this answer? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about A heavy hollow cone of radius R and height h is placed on a horizontal table surface, with its flat base on the table. The whole volume inside the cone is filled with water of density r. The circular rim of the cone's base has a watertight seal with the table's surface and the top apex of the cone has a small hole. Neglecting atmospheric pressure find the total upward force exerted by water on the cone isa)(2/3)pR2hrgb)(1/3)pR2hrgc)pR2hrgd)NoneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A heavy hollow cone of radius R and height h is placed on a horizontal table surface, with its flat base on the table. The whole volume inside the cone is filled with water of density r. The circular rim of the cone's base has a watertight seal with the table's surface and the top apex of the cone has a small hole. Neglecting atmospheric pressure find the total upward force exerted by water on the cone isa)(2/3)pR2hrgb)(1/3)pR2hrgc)pR2hrgd)NoneCorrect answer is option 'A'. Can you explain this answer?.
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