Pascals law FLUID MECHANICS

2. Pascal’s Law
According to Pascal’s Law, Pressure or intensity of pressure at a point in a static fluid will be
equal in all directions.
Let us consider one arbitrary fluid element of rectangular shape ABC as displayed here in following
figure. Let us assume that width of fluid element ABC perpendicular to the plane of paper is unity.
Let us consider the following terms as mentioned here
PX = Pressure acting in X- direction over the face AB
PY = Pressure acting in Y- direction over the face AC
PZ = Pressure acting in Z- direction over the face BC
θ = Angle ABC, as displayed above in figure
dx, dy and ds : Fluid element dimensions
ρ = Density of the fluid

3. Let us analyse here the forces acting on the fluid
element ABC
Force on the face AB, AC and BC
FAB = PX x Area of face AB = PX. dy. 1 = PX. dy
FAC = PY x Area of face AC = PY. dx. 1 = PY. dx
FBC = PZ x Area of face BC = PZ. ds. 1 = PZ. ds
Weight of the fluid element,
W = Volume x Density of fluid x acceleration due to
gravity
W = Area x width of fluid element x Density of fluid
x acceleration due to gravity
W = (AB x AC/2) x 1 x ρ x g = (dy dx/2) x ρ x g

4. Considering the forces in X-direction
Px. dy- PZ. ds Sin (90- θ) = 0
PX. dy = PZ. ds Cos θ
As we can see from above fluid element ABC, dy = ds Cos θ
PX. dy = PZ. dy
PX = PZ............................................(1)
Considering the forces in Y-direction
PY. dx - PZ. ds Cos (90- θ) - (dy dx/2) x ρ x g = 0
PY. dx - PZ. ds Sin θ - (dy dx/2) x ρ x g = 0
As fluid element is very small and therefore, we can neglect
the weight of fluid element
PY. dx - PZ. ds Sin θ = 0
As we can see from above fluid element ABC, dx = ds Sin θ
PY. dx - PZ dx = 0
PY = PZ................................................(2)

5. From last two expressions mentioned in eq
(1) AND eq(2), we can write following
equation as mentioned here
PX = PY = PZ.............................prooved
We can say from above equation that pressure at
any point in X, Y and Z directions will be same.
Pascal’s Law provides the base for any hydraulic
system or we can say that complete hydraulic
system is based on the principle of Pascal’s Law.

6. Let us understand, how hydraulic system is based on Pascal's Law
As we know that pressure at every point in enclosed liquid will be same
and hence there is no matter about the shape of vessel or container in
which liquid is placed.In order to understand how hydraulic system
depends over Pascal’s law , we will consider following case.
P1= F1/A1
And
P2= F2/A2
According to Pascal's law
P1= P2
F1/A1 = F2/A2
F1 =F2 [A1/A2]
As we may see in above figure, area A2 is larger as compared to area A1
hence we will require less force to lift the heavy load.
This is the basic principle which is used by all hydraulic system. For more
detailed information about the Pascal's Law, we must have to find the post
i.e. Application of fluid power: Hydraulic Jack.