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A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure?
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A plane area in the form of a right angle triangle height h is immerse...
Total Force on a Triangular Plane
To calculate the total force exerted by water on one side of a right triangle submerged vertically, we use hydrostatic principles. The pressure at a depth \( y \) is given by:
\[ P = \rho g y \]
where:
- \( \rho \) = density of water (approximately \( 1000 \, \text{kg/m}^3 \))
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( y \) = depth below the water surface
The total force \( F \) acting on the triangular plane can be computed by integrating the pressure over the area of the triangle.
Force Calculation
- The area \( A \) of the triangle is:
\[ A = \frac{1}{2} \times b \times h \]
where \( b \) is the base and \( h \) is the height.
- The total force \( F \) is obtained by integrating the pressure over the area:
\[ F = \int_0^h P \, dA \]
- The average pressure over the triangle can be calculated as:
\[ \bar{P} = \frac{\rho g h}{2} \]
- Thus, the total force can be expressed as:
\[ F = \bar{P} \times A = \frac{\rho g h}{2} \times \frac{1}{2} b h = \frac{1}{4} \rho g b h^2 \]
Location of Centre of Pressure
The center of pressure \( y_{cp} \) can be calculated using the formula:
\[ y_{cp} = \frac{I_G}{A \bar{y}} + \bar{y} \]
where:
- \( I_G \) = moment of inertia of the triangle about the centroid
- \( \bar{y} = \frac{h}{3} \) (depth of the centroid from the surface)
The moment of inertia for a right triangle is:
\[ I_G = \frac{b h^3}{36} \]
Substituting these values gives:
\[ y_{cp} = \frac{\frac{b h^3}{36}}{\frac{1}{2} b h \cdot \frac{h}{3}} + \frac{h}{3} \]
This yields the final position of the center of pressure below the surface.
To calculate the total force exerted by water on one side of a right triangle submerged vertically, we use hydrostatic principles. The pressure at a depth \( y \) is given by:
\[ P = \rho g y \]
where:
- \( \rho \) = density of water (approximately \( 1000 \, \text{kg/m}^3 \))
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( y \) = depth below the water surface
The total force \( F \) acting on the triangular plane can be computed by integrating the pressure over the area of the triangle.
Force Calculation
- The area \( A \) of the triangle is:
\[ A = \frac{1}{2} \times b \times h \]
where \( b \) is the base and \( h \) is the height.
- The total force \( F \) is obtained by integrating the pressure over the area:
\[ F = \int_0^h P \, dA \]
- The average pressure over the triangle can be calculated as:
\[ \bar{P} = \frac{\rho g h}{2} \]
- Thus, the total force can be expressed as:
\[ F = \bar{P} \times A = \frac{\rho g h}{2} \times \frac{1}{2} b h = \frac{1}{4} \rho g b h^2 \]
Location of Centre of Pressure
The center of pressure \( y_{cp} \) can be calculated using the formula:
\[ y_{cp} = \frac{I_G}{A \bar{y}} + \bar{y} \]
where:
- \( I_G \) = moment of inertia of the triangle about the centroid
- \( \bar{y} = \frac{h}{3} \) (depth of the centroid from the surface)
The moment of inertia for a right triangle is:
\[ I_G = \frac{b h^3}{36} \]
Substituting these values gives:
\[ y_{cp} = \frac{\frac{b h^3}{36}}{\frac{1}{2} b h \cdot \frac{h}{3}} + \frac{h}{3} \]
This yields the final position of the center of pressure below the surface.
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A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure?
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A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure?.
A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A plane area in the form of a right angle triangle height h is immersed vertically in water with its vertex at the water surface. Calculate the total force on one side of the triangular plane and the location of the centre of pressure?.
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