Short Notes: Forces on Submerged Surface | Short Notes for Mechanical Engineering PDF Download

 Page 1


Forces on Submerged Surface 
Fluid Mechanics is high scoring subject in all Mechanical Engineering exams such 
as GATE, ESE, ISRO and other PSU exams.After going through Manometry and 
Buoyancy chapter. Forces on Submerged Surfaces in a static fluid is also an 
important section. It deals with Horizontal, vertical and curved surfaces submered 
under fluid and the various forces applied on these section. Some easy, which 
other tricky questions are expected from this section.
Forces on Submerged Surfaces in Static Fluids
• Pressures at any equal depths in a continuous fluid are equal
• Pressure at a point acts equally in all directions (Pascal’s law).
• Forces from a fluid on a boundary acts at right angles to that boundary.
General Submerged Plane
• The total or resultant force, R , on the plane is the sum of the forces on the 
small elements i.e.
R = p,<S4, + p2 SA2+...... pnSAn = £ pSA
and the resultant force will act through the centre of pressure.
Page 2


Forces on Submerged Surface 
Fluid Mechanics is high scoring subject in all Mechanical Engineering exams such 
as GATE, ESE, ISRO and other PSU exams.After going through Manometry and 
Buoyancy chapter. Forces on Submerged Surfaces in a static fluid is also an 
important section. It deals with Horizontal, vertical and curved surfaces submered 
under fluid and the various forces applied on these section. Some easy, which 
other tricky questions are expected from this section.
Forces on Submerged Surfaces in Static Fluids
• Pressures at any equal depths in a continuous fluid are equal
• Pressure at a point acts equally in all directions (Pascal’s law).
• Forces from a fluid on a boundary acts at right angles to that boundary.
General Submerged Plane
• The total or resultant force, R , on the plane is the sum of the forces on the 
small elements i.e.
R = p,<S4, + p2 SA2+...... pnSAn = £ pSA
and the resultant force will act through the centre of pressure.
Note: For a plane surface all forces acting can be represented by one single
resultant force, acting at right-angles to the plane through the centre of pressure.
Horizontal Submerged plane
• The pressure, p, will be equal to all points of the surface.
• The resultant force will be given by R = pressure area of plane = pA
Curved Submerged Surface
• Each elemental force is a different magnitude and in a different direction (but 
still normal to the surface.).
• It is, in general, not easy to calculate the resultant force for a curved surface 
by combining all elemental forces.
• The sum of all the forces on each element willalwaysbe less than the sum of 
the individual forces, £pA
Resultant Force and Centre of Pressure on a general plane surface in a liquid
• Take pressure as zero at the surface.
• Measuring down from the surface, the pressure on an element 3A, depth z, p = 
pgz
• So force on element F = pgzdA
• Resultant force on plane
R=pg£z3A
• £zdA is known as the1 st Moment of Area of the plane PQ about the free 
surface.
And it is known that
y zSA = Az
• A is the area of the plane,z is the distance to the centre of gravity (centroid).
• In terms of distance from point 0, £z3A= AxsinG (= 1st moment of area about 
a line through 0 x sin©)
• The moment of R will be equal to the sum of the moments of the forces on all 
the elements G A about the same point.
It is convenient to take moment about 0.
• The force on each elemental area:
Page 3


Forces on Submerged Surface 
Fluid Mechanics is high scoring subject in all Mechanical Engineering exams such 
as GATE, ESE, ISRO and other PSU exams.After going through Manometry and 
Buoyancy chapter. Forces on Submerged Surfaces in a static fluid is also an 
important section. It deals with Horizontal, vertical and curved surfaces submered 
under fluid and the various forces applied on these section. Some easy, which 
other tricky questions are expected from this section.
Forces on Submerged Surfaces in Static Fluids
• Pressures at any equal depths in a continuous fluid are equal
• Pressure at a point acts equally in all directions (Pascal’s law).
• Forces from a fluid on a boundary acts at right angles to that boundary.
General Submerged Plane
• The total or resultant force, R , on the plane is the sum of the forces on the 
small elements i.e.
R = p,<S4, + p2 SA2+...... pnSAn = £ pSA
and the resultant force will act through the centre of pressure.
Note: For a plane surface all forces acting can be represented by one single
resultant force, acting at right-angles to the plane through the centre of pressure.
Horizontal Submerged plane
• The pressure, p, will be equal to all points of the surface.
• The resultant force will be given by R = pressure area of plane = pA
Curved Submerged Surface
• Each elemental force is a different magnitude and in a different direction (but 
still normal to the surface.).
• It is, in general, not easy to calculate the resultant force for a curved surface 
by combining all elemental forces.
• The sum of all the forces on each element willalwaysbe less than the sum of 
the individual forces, £pA
Resultant Force and Centre of Pressure on a general plane surface in a liquid
• Take pressure as zero at the surface.
• Measuring down from the surface, the pressure on an element 3A, depth z, p = 
pgz
• So force on element F = pgzdA
• Resultant force on plane
R=pg£z3A
• £zdA is known as the1 st Moment of Area of the plane PQ about the free 
surface.
And it is known that
y zSA = Az
• A is the area of the plane,z is the distance to the centre of gravity (centroid).
• In terms of distance from point 0, £z3A= AxsinG (= 1st moment of area about 
a line through 0 x sin©)
• The moment of R will be equal to the sum of the moments of the forces on all 
the elements G A about the same point.
It is convenient to take moment about 0.
• The force on each elemental area:
Force on SA = pgzSA
- sin 9SA
• The moment of this force is:
Moment of Force on SA about O = /C ^.vsin OSA x s
= pg sin 9 SAs2
p,g and 0 are the same for each element, giving the total moment as 
Sum of moments = pg sin 0^ s 2SA
Moment of R about O = R x Sc = pgAx sin 9SC
Equating
pgAx sin 9 Sc = pg sin 9^ s2 SA
• The position of the centre of pressure along the plane measure from the point
0 is:
Y s 2 SA
¦>
2nd moment of area about O = I0 = 2^s SA
• The position of the centre of pressure along the plane measure from the point
0 is:
° Sc = 2nd Moment of area about a line through 0/1 st Moment of area 
about a line through 0 
° Depth to the centre of pressure =Sc sin0
2nd moment of area is a geometric property, about a line through the centroid of 
some common shapes is given below:
Page 4


Forces on Submerged Surface 
Fluid Mechanics is high scoring subject in all Mechanical Engineering exams such 
as GATE, ESE, ISRO and other PSU exams.After going through Manometry and 
Buoyancy chapter. Forces on Submerged Surfaces in a static fluid is also an 
important section. It deals with Horizontal, vertical and curved surfaces submered 
under fluid and the various forces applied on these section. Some easy, which 
other tricky questions are expected from this section.
Forces on Submerged Surfaces in Static Fluids
• Pressures at any equal depths in a continuous fluid are equal
• Pressure at a point acts equally in all directions (Pascal’s law).
• Forces from a fluid on a boundary acts at right angles to that boundary.
General Submerged Plane
• The total or resultant force, R , on the plane is the sum of the forces on the 
small elements i.e.
R = p,<S4, + p2 SA2+...... pnSAn = £ pSA
and the resultant force will act through the centre of pressure.
Note: For a plane surface all forces acting can be represented by one single
resultant force, acting at right-angles to the plane through the centre of pressure.
Horizontal Submerged plane
• The pressure, p, will be equal to all points of the surface.
• The resultant force will be given by R = pressure area of plane = pA
Curved Submerged Surface
• Each elemental force is a different magnitude and in a different direction (but 
still normal to the surface.).
• It is, in general, not easy to calculate the resultant force for a curved surface 
by combining all elemental forces.
• The sum of all the forces on each element willalwaysbe less than the sum of 
the individual forces, £pA
Resultant Force and Centre of Pressure on a general plane surface in a liquid
• Take pressure as zero at the surface.
• Measuring down from the surface, the pressure on an element 3A, depth z, p = 
pgz
• So force on element F = pgzdA
• Resultant force on plane
R=pg£z3A
• £zdA is known as the1 st Moment of Area of the plane PQ about the free 
surface.
And it is known that
y zSA = Az
• A is the area of the plane,z is the distance to the centre of gravity (centroid).
• In terms of distance from point 0, £z3A= AxsinG (= 1st moment of area about 
a line through 0 x sin©)
• The moment of R will be equal to the sum of the moments of the forces on all 
the elements G A about the same point.
It is convenient to take moment about 0.
• The force on each elemental area:
Force on SA = pgzSA
- sin 9SA
• The moment of this force is:
Moment of Force on SA about O = /C ^.vsin OSA x s
= pg sin 9 SAs2
p,g and 0 are the same for each element, giving the total moment as 
Sum of moments = pg sin 0^ s 2SA
Moment of R about O = R x Sc = pgAx sin 9SC
Equating
pgAx sin 9 Sc = pg sin 9^ s2 SA
• The position of the centre of pressure along the plane measure from the point
0 is:
Y s 2 SA
¦>
2nd moment of area about O = I0 = 2^s SA
• The position of the centre of pressure along the plane measure from the point
0 is:
° Sc = 2nd Moment of area about a line through 0/1 st Moment of area 
about a line through 0 
° Depth to the centre of pressure =Sc sin0
2nd moment of area is a geometric property, about a line through the centroid of 
some common shapes is given below:
Shape Area A
2n d moment of area, 
about
an axis through the centroid
Rectangle
b
b d
b d 3 
12
L
- G
Triangle
bd
b d 3
2
36 b
Circle
A ? Y _ c
k R a
V 7
4
Semicircle
G ~ r
x R 2
n 11 m
. ,x _ a . . A . J e h 'V* 2
U*I IUZK
Submerged vertical surface
• Pressure diagram Method
° For vertical walls of constant width it is possible to find the resultant 
force and centre of pressure graphically using a pressure diagram.
° We know the relationship between pressure and depth:p =pgz
• So we can draw the diagram below:
• This is known as a pressure diagram.
• Pressure increases from zero at the surface linearly by p = pgz, to a maximum 
at the base of p = pgH.
• The area of this triangle represents the resultant force per unit width on the 
vertical wall
Area = - x AB x BC 
2
= \H p g H
= \ p s U 2
• Resultant force per unit width : R= 1/2 pgH2
Page 5


Forces on Submerged Surface 
Fluid Mechanics is high scoring subject in all Mechanical Engineering exams such 
as GATE, ESE, ISRO and other PSU exams.After going through Manometry and 
Buoyancy chapter. Forces on Submerged Surfaces in a static fluid is also an 
important section. It deals with Horizontal, vertical and curved surfaces submered 
under fluid and the various forces applied on these section. Some easy, which 
other tricky questions are expected from this section.
Forces on Submerged Surfaces in Static Fluids
• Pressures at any equal depths in a continuous fluid are equal
• Pressure at a point acts equally in all directions (Pascal’s law).
• Forces from a fluid on a boundary acts at right angles to that boundary.
General Submerged Plane
• The total or resultant force, R , on the plane is the sum of the forces on the 
small elements i.e.
R = p,<S4, + p2 SA2+...... pnSAn = £ pSA
and the resultant force will act through the centre of pressure.
Note: For a plane surface all forces acting can be represented by one single
resultant force, acting at right-angles to the plane through the centre of pressure.
Horizontal Submerged plane
• The pressure, p, will be equal to all points of the surface.
• The resultant force will be given by R = pressure area of plane = pA
Curved Submerged Surface
• Each elemental force is a different magnitude and in a different direction (but 
still normal to the surface.).
• It is, in general, not easy to calculate the resultant force for a curved surface 
by combining all elemental forces.
• The sum of all the forces on each element willalwaysbe less than the sum of 
the individual forces, £pA
Resultant Force and Centre of Pressure on a general plane surface in a liquid
• Take pressure as zero at the surface.
• Measuring down from the surface, the pressure on an element 3A, depth z, p = 
pgz
• So force on element F = pgzdA
• Resultant force on plane
R=pg£z3A
• £zdA is known as the1 st Moment of Area of the plane PQ about the free 
surface.
And it is known that
y zSA = Az
• A is the area of the plane,z is the distance to the centre of gravity (centroid).
• In terms of distance from point 0, £z3A= AxsinG (= 1st moment of area about 
a line through 0 x sin©)
• The moment of R will be equal to the sum of the moments of the forces on all 
the elements G A about the same point.
It is convenient to take moment about 0.
• The force on each elemental area:
Force on SA = pgzSA
- sin 9SA
• The moment of this force is:
Moment of Force on SA about O = /C ^.vsin OSA x s
= pg sin 9 SAs2
p,g and 0 are the same for each element, giving the total moment as 
Sum of moments = pg sin 0^ s 2SA
Moment of R about O = R x Sc = pgAx sin 9SC
Equating
pgAx sin 9 Sc = pg sin 9^ s2 SA
• The position of the centre of pressure along the plane measure from the point
0 is:
Y s 2 SA
¦>
2nd moment of area about O = I0 = 2^s SA
• The position of the centre of pressure along the plane measure from the point
0 is:
° Sc = 2nd Moment of area about a line through 0/1 st Moment of area 
about a line through 0 
° Depth to the centre of pressure =Sc sin0
2nd moment of area is a geometric property, about a line through the centroid of 
some common shapes is given below:
Shape Area A
2n d moment of area, 
about
an axis through the centroid
Rectangle
b
b d
b d 3 
12
L
- G
Triangle
bd
b d 3
2
36 b
Circle
A ? Y _ c
k R a
V 7
4
Semicircle
G ~ r
x R 2
n 11 m
. ,x _ a . . A . J e h 'V* 2
U*I IUZK
Submerged vertical surface
• Pressure diagram Method
° For vertical walls of constant width it is possible to find the resultant 
force and centre of pressure graphically using a pressure diagram.
° We know the relationship between pressure and depth:p =pgz
• So we can draw the diagram below:
• This is known as a pressure diagram.
• Pressure increases from zero at the surface linearly by p = pgz, to a maximum 
at the base of p = pgH.
• The area of this triangle represents the resultant force per unit width on the 
vertical wall
Area = - x AB x BC 
2
= \H p g H
= \ p s U 2
• Resultant force per unit width : R= 1/2 pgH2
Note: For a triangle the centroid is at 2/3 its height i.e. the resultant force acts 
horizontally through the point z= 2/3 H
Checking against Moment method
• The resultant force is given by:
R = pgAz = pgAx sin 0 
= pg(H x l ) y sin#
• and the depth to the centre of pressure by:
D = sin
• and by the parallel axis theorem (with width of 1)
Jo = I GG + A^ 2
j x j I + V x 4 k ) 2 -_h L
12 \ 2 J 3 •
• Depth to the centre of pressure
D
' i 3^ 
vH 2 I2 j
2
3
H
Submerged Curved Surface
• If the surface is curved the resultant force must be found by combining the 
elemental forces using some vectorial method.
• Calculate the horizontal and vertical components and combine these to 
obtain the resultant force and direction.
Consider the Horizontal forces